What in nature causes Newton's gravitation constant to have its given value? Does the value of Newton's universal gravitational constant $G$ remain a mystery? Why does it have the value that it has?
 A: This video by Richard P. Feynman might explain how hard it is to answer such a 'why' question. An excerpt:

But the problem, you see, when you ask why something happens, how does a person answer why something happens? ... When you explain a why, you have to be in some framework that you allow something to be true. You have to know what it is that you’re permitted to understand and allow to be understood and known, and what it is you’re not. ...So I am not going to be able to give you an answer to [the question, in this case why is the value of G what it is] except to tell you that it is.

The value of 'big $G$' is something that we can measure to incredible precision, but I doubt that there is any meaningful answer to the question 'why' it has the value it has. There are some things you just have to accept as true, and in this case I would believe that 'big $G$'s value is something that has to simply be accepted. It's a good thing it is what it is, and there are some nice discussions as to what would happen if it took on a different value. 
A: The value of the gravitational constant (aka Newton's constant and big G) is a man-made convention - it's dimensionful and depends upon the our definition of units. In common SI units,
$$
G = 6.674\times 10^{-11}\,\text{Nm$^2$/kg$^2$},
$$
but clearly the numerical value would differ in e.g. units of $\text{N miles$^2$/stones$^2$}$.
With that in mind, it makes no sense to attempt to derive the value of big G and little sense to claim that it's value is a mystery.
The most convenient choice if units is so-called natural units aka Planck units, in which, inter alia, one sets 
$$G=1$$
There's nothing to explain about the arbitrary value of one dimensionful number. Dimensionless ratios of numbers, on the other hand, are indeed mysterious. Indeed, the real mystery is why the other forces are so much stronger than gravity. 
The wiki page for Planck units, helped by Frank Wliczek, makes this point rather well, so I quote it below:

Planck units are free of anthropocentric arbitrariness ... Unlike the metre and second, which exist as base units in the SI system for historical reasons, the Planck length and Planck time are conceptually linked at a fundamental physical level.
Natural units help physicists to reframe questions. Frank Wilczek puts it succinctly:
We see that the question [posed] is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number [1/(13 quintillion)].
... the electrostatic repulsive force between two protons (alone in free space) greatly exceeds the gravitational attractive force between the same two protons ... the disparity of magnitude of force is a manifestation of the fact that the charge on the protons is approximately the unit charge but the mass of the protons is far less than the unit mass.

A: I can provide only "classically newtonian" answer cause my orientation in general relativity is only very limited.
As it's written in the wikipedia article, the $G$ is empirical constant. How to understand that? Experimental scientists are usually aware of dependency on parameters, but the absolute values are corrected using various additive or (in this case) multiplicative constants. They knew that gravitational forces are proportional to masses of interacting objects and inverse square law. What is left is to measure the constant which would make the general dependency an equality.
A: Newton just find that the gravitational force is proportional to the product of mass and the inverse of the square of the distance.
Some years after, Cavendish mesure the value of that constant G. That constan could be 1, or  1000000.  But G=6.674×10−11 N · (m/kg)2. Why?, nature works that way.
