Relativity of Simultaniety 
In the first figure, A and B are two equidistant points from the observer O in S. In the second figure (reference frame S') the corresponding points are A'and B' such that A'O'=O'B', where O' is the observer in S'.
The first figure represents the state of S and S' at time t=0 when two events occur at A and B in S and the corresponding points In S' are A' and B'. Now S' moves with a velocity v w.rt. S. 
The second figure represents the state of S and S' at time t=t1 when the rays from A and B reaches O simultaneously (if possible). Clearly, since at that time O and O' don't coincide, to O' the events should not be simultaneous.
Conversely, if at time t=t1' the rays from A' and B' reaches O' simultaneously (if possible), for the same reason O will not find the events to be simultaneous.
Since the light rays are unique, they cannot meet more than once. So only one observer should find the the events to be simultaneous.
In this case which observer will find the events to be simultaneous and why?
 A: 
In this case which observer will find the events to be simultaneous
  and why?

Briefly, the events are simultaneous in $S$ and the reason is that you've stipulated the events are simultaneous in $S$.
In more detail...

when two events occur at A and B in S and the corresponding points In
  S' are A' and B'.

The wording here is puzzling.  Events don't occur, events are 'points' in spacetime - an event simply is.
The primed and unprimed coordinate systems assign, in general, different space and time coordinates to events but the events are fundamental.
As best as I can tell, you're asking in which frame two particular events are observed to be simultaneous.
And, evidently, the two events, which have space-like interval, are the emission of oppositely directly light rays 
But, as drawn and as stipulated, the two events are simultaneous in $S$.

at time t=0 when two events occur at A and B in S

Do you see?  There is no question as to whether the events are simultaneous in $S$ since you've stipulated that the events occur when $t=0$ where, I assume, $t$ is the coordinate time in the unprimed frame $S$.
Thus, the events $A$ and $B$ have the same time coordinate in $S$ which means they are simultaneous in $S$.
But, the two events aren't simultaneous in $S'$ since the primed frame has velocity $v$ in $S$ and thus, as is well known, the two events will have differing $t'$ coordinates as per the Lorentz transformation.
A: Let us go the extreme case where o=o at t=0.
If you follow the equations, there is only one observer who can perceive the events as simultaneous,  the other one will perceive it as being not simultaneous. 
Just let's go to the classical example. A guy in the middle of a train turns on two lanterns in opposite directions.  He will perceive the time when the two end of the wagon are illuminated as simultaneous (in his reference frame c is a constant). Somebody outside the wagon will see the back being illuminated before the front (not simultaneous).  This fact will not change in any other reference frame (they will always agree that (1) perceives it as simultaneous and (2) does not)
A: If you suppose that simultaneity arises in the $S$ frame, this means that the emission of the rays correspond to the following space-time events , expressed in coordinates $(x,t)$ in the $S$ frame : 
$$A(-l,0)\quad B(l,0)$$ 
Simultaneity means that the time coordinates $t$ of $A$ and $B$ are equal.
Now, these same space-time events, expressed in the $x'$, $t'$ coordinates of the $S'$ frame, thanks to the Lorentz transformations, $x'= \gamma(v)(x-vt)$ and $t'= \gamma(v)(t- \frac{vx}{c^2})$, are: 
$$A'(-l \gamma(v), \gamma(v) \frac{vl}{c^2}) \quad B'(l \gamma(v), -\gamma(v) \frac{vl}{c^2}), $$
where
$$\gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}.$$
It appears clearly that the times coordinate $t'$  of the space-time events $A'$ and  $B'$ are not equals, so they do not appear simultaneously, relatively to the $S'$ frame. 
A: In response to the extended discussion in the comments I wrote a program to clarify things interactively. The program is here:
https://www.khanacademy.org/cs/relativistic2/6050744190369792
Frame S:

The three perfectly vertical lines are the points A, O, and B being stationary. The other three lines are A', O', and B'. In frame S, the flashes of light at t=0 on lines A and B hit O at the same time. But O' moves into the flash of light from B first, and the flash of light from A catches up and hits it later. So the flash of light from line B at t=0 hits O' first.
Frame S':

Now the lines A', O', and B' are stationary (perfectly vertical).The flash from line B still hits O' first, but now we see that in S', flash B started before flash A. But, the flashes from B and A still hit line O at the exact same time. So there is no contradiction. These two view points give rise to exactly the same physical situation.
A: 
In the first figure 

... actually: in the bottom parts of both the first and the second figures ...

A and B are two equidistant points from the observer O in S. 

In detail, therefore


*

*A, O, and B were and remained (pairwise) at rest to each other, with O having been and remained the "middle between" A and B,

*for each signal indication stated by O, the corresponding reflection indications of A and of B were in turn observed by O in coincidence,

*for each signal indication stated by A, the corresponding reflection indications of B was in turn observed by A in coincidence with the indication of O having observed the indication of A having observed the indication of O reflecting the signal indication by A, and

*for each signal indication stated by B, the corresponding reflection indications of A was in turn observed by B in coincidence with the indication of O having observed the indication of B having observed the indication of O reflecting the signal indication by B.

In the second figure (reference frame S') 

... actually: in the top parts of both the first and the second figures ...

the corresponding points are A'and B' such that A'O'=O'B', where O' is the observer in S'.

In detail, therefore


*

*A', O', and B' were and remained (pairwise) at rest to each other, with O' having been and remained the "middle between" A' and B',

*for each signal indication stated by O', the corresponding reflection indications of A' and of B' were in turn observed by O' in coincidence,

*for each signal indication stated by A', the corresponding reflection indications of B' was in turn observed by A' in coincidence with the indication of O' having observed the indication of A' having observed the indication of O reflecting the signal indication by A', and

*for each signal indication stated by B', the corresponding reflection indications of A' was in turn observed by B' in coincidence with the indication of O' having observed the indication of B' having observed the indication of O' reflecting the signal indication by B'.

The first figure represents the state of S and S' at time t=0 

The first figure shows the meetings (coincidence events) of A and A', of O and O', and of B and B'.
But surely it is completely insignificant to physics and geometry whether and how the participants, their individual indications, or the events in which they (pairwise) participated might be sprinkled with coordinate numbers.

when the two events occur at A and B in S and the corresponding points In S' are A' and B'. 

Does the word "when" in this description of the first figure already stand for some particular setup condition involving some determination(s) of simultaneity? 
Do you mean to show, perhaps, that 


*

*the indication by A of having met (and having been passed) by A' had been simultaneous to

*the indication by O of having met (and having been passed) by O', as well as simultaneous to

*the indication by B of having met (and having been passed) by B' ?
If so, then answers to the subsequent questions are thereby already anticipated.
If not, then this instance of using the word "when" is superfluous.

Now S' moves with a velocity v w.rt. S. The second figure represents [...] the rays from A and B reaches O simultaneously (if possible). 

The better, unambiguous terminology is to entertain the possibility that the rays from A and B reached O in coincidence. 
Yes, that is possible; i.e. it is consistent with the setup description so far, including the instance of using the word "when" noted above.
Importantly:
It is of course straightforward (at least in principle) for O to make the experimental determination whether O observed the event of A and A' having met and passed each other in coincidence with with the event of B and B' having met and passed each other; or not.
If so, then, by Einstein's (coordinate-free) definition of how to determine "simultaneity",  the indication by A of having met (and having been passed) by A' is said to have been simultaneous to the indication by B of having met (and having been passed) by B'. (And if not, then not.)     
Moreover: if so, then it can be concluded that the indication by A' of having met (and having been passed) by A is said to have been not simultaneous to the indication by B' of having met (and having been passed) by B;
since O', who as the "middle between" A' and B' is decisive in making the determination, would have observed the event of B and B' having met and passed each other not in coincidence with with the event of A and A' having met and passed each other; but O' first observed the former, and only subsequently the latter.

Conversely, if [...] the rays from A' and B' reaches O' [in coincidence] (if possible)

This, itself, is also consistent with the above setup description; at least if the noted instance of the word "when" is dropped.
Yes, instead of the case considered above, O' could find having observed the event of A and A' having met and passed each other in coincidence with with the event of B and B' having met and passed each other;
and consequently the indication by A' of having met (and having been passed) by A would be said to have been simultaneous to the indication by B' of having met (and having been passed) by B.
But obviously this would be the converse of, and mutually exclusive to, the other other case described eariler.
It is even consistent with the above setup description, at least if the noted instance of the word "when" is dropped, that neither O nor O' would find having observed the event of A and A' having met and passed each other in coincidence with with the event of B and B' having met and passed each other.

[...] which observer will find the events to be simultaneous and why? 

Simultaneity, as determined by Einstein's definition, does not apply to entire events, but only (if at all) to indications of some individual participants. Participant O is (by the setup description) "middle between" participants A and B, and therefore decisive in determining simultaneity (or sequence) of the indications of A and of B. 
But obviously O is not "middle between" participants A' and B'. Instead, the determination of simultaneity (or sequence) of the indications of A' and of B' is up to the participant who is identified as "middle between" participants A' and B', namely O'.
Vice versa, O' is not "middle between" participants A and B; the determination of simultaneity (or sequence) of the indications of A and of B is up to the participant who is identified as "middle between" participants A and B, namely O.
A: According to Special Relativity:
            Given, at time t=0
AO=OB=A'O'=O'B'.
To the observer O, since AO=OB, the rays from A and B will reach O at the same time and hence, O will find the events to be simultaneous. Furthermore, since, at t=0 (the instant when the events occurred) A coincides with A' and B with B' and the speed of light is absolute, the rays from A' and B' will also meet at O. So for O a single will come from A and A' and another single ray will come from B and B' with the same speed and hence will meet at the midpoint O such that O will observe the events to be simultaneous.
Since, the laws of physics are same for all inertial observers, O' will also find the events to be simultaneous in a similar way.
So, according to SR both the observers will find the events to be simultaneous.
But the fact is, due to relative motion between S and S' at time t>0, O and O' can't coincide unless the speed of light be infinity.
Therefore, for a finite speed of light, SR will be true iff there will be two different rays coming from A and A' with different speeds. However, this possibility is not permitted by the second postulate of SR itself.
The above analysis leads to a contradictory situation such that either such event is an impossible event or SR is WRONG.
According to Newtonian Relativity:
              In the frame S (consider it as the absolute frame), O will observe the events simultaneously for the rays from A and B. The corresponding rays from A' and B' will also meet at O with speed c+v and c-v respectively.
In the frame S', since it is moving with a velocity v w.r.t. S, the rays from A' and B' will meet at the corresponding vertical point of O in S' with speed c+v and c-v respectively while the rays from A and B will meet at the same point with speed c.
The choice of absolute frame here is arbitrary. It only let you to consider that the speed of light in the absolute frame is c.
Through this experiment, the observer can be able to know whether he is in the absolute frame or in another inertial frame. If the observer finds his observation similar to O i.e. observes the events to be simultaneous at his position, he is in the rest frame and otherwise he is in a moving frame.
A: As far as I understand your experiment, A and B, as well as A' and B' should actually be considered sources of light. 
Now, if A and B are co-moving with O (and therefore, all three are stationary wrt. each other), the rays from A and B will reach the observer O at the same time. However, O and A' are moving toward each other, and therefore the observer O will intercept the ray from A' sooner than the ray from B' will reach him. 
The situation is reversed in the case of O'.
So, both observers will see light coming from the two sources in their respective frames of reference simultaneously, while the light coming from the sources located in the other frame will reach them desynchronized.
