Relativity of Simultaneous Events I'm used to relativity of simultaneity situations that are like the Einstein lightning train thought experiment. This problem is different. Say an observer on the space station observes lamps 1 and 2 blinking at the same time. According to an observer on the rocket, lamp 2 goes off first. Why is this? 

 A: This problem isn't different. This is the classic illustration of the relativity of simultaneity. 
There are pretty many pretty ways to prove that if two events are spatially separated and happen simultaneously with respect to one frame then they wouldn't be simultaneous with respect to any other inertial frame that is moving with respect to the first one. The neat way of seeing this would be via recalling the Lorentz transformations which tell us how space and time intervals between the same events, as observed in one frame, relate to those observed in the other.
So, the time interval between a pair of events as observed by the rocket, say $\Delta t'$, relates to the spatial interval $\Delta x$ and the time interval $\Delta t$ between the same events in the space station frame by $$\Delta t'=\frac{\Delta t-\frac{v}{c^2}\Delta x}{\sqrt{1-\frac{v^2}{c^2}}}$$where $v$ is the velocity of the rocket with respect to the space station. In your example, since the blinks are simultaneous in the space station frame, $\Delta t=0$. And the thus, $\text{Sign}(\Delta t')=\text{Sign}(-v\Delta x)$. In more explicit terms, $\text{Sign}(t_2'-t_1')=-\text{Sign}(v(x_2-x_1))$. Thus, in the space station frame, the event who is situated towards the direction of the velocity of the rocket compared to the location of the other event, will, in the rocket's frame, happen before the other event. In other words, if the relative position of an event (say event $2$) with respect to the other event (so event $1$), i.e., $x_2-x_1$, has the same direction as the direction of the velocity $v$ (all in the space station frame), then, in the rocket frame, event $2$ will happen before event $1$.  So, that is why lamp $2$ will blink before lamp $1$ blinks in the frame of the rocket. 
A: Even the observer on the station only observes them to flash simultaneously if he is equidistant from both. 
The rocket observer on that trajectory is never equidistant from both except at a single instant as he passes. 
But because observation of an electromagnetic signal is not instantaneous and does not take place in an instant, and because the rocket position evolves in time during the process of observation (because it is moving), and because the Doppler effect changes the observed wavelength of both signals (non-symmetrically - at the point when the rocket is equidistant, one is red-shifted and the other blue-shifted, and thus it takes a different amount of time to observe one cycle), there is no way to time the observation process on the rocket such that both signals are received at the same time.
Relativity provides the tool with which we determine the order and timing at which the signals will actually be observed on the rocket.
