Is there a name for what Feynman called a fundamental constant i.e. "ratio of electrical repulsion to gravitational attraction between electrons"? Paraphrasing from Feynman's lecture on physics, from the chapter on gravity

If we take, in some natural units, the repulsion of two electrons (nature’s universal charge) due to electricity, and the attraction of two electrons due to their masses, we can measure the ratio of electrical repulsion to the gravitational attraction. The ratio is independent of the distance and is a fundamental constant of nature. The ratio is shown in Fig. 7–14. The gravitational attraction relative to the electrical repulsion between two electrons is $1 / 4.17×10^{42}$! The question is, where does such a large number come from? It is not accidental, like the ratio of the volume of the earth to the volume of a flea.

Does this fundamental constant of nature have a name? And does it have some profound significance in physics? Is it another one of those "fine-tuned " constants? Or would a slight change in it  not matter much?
 A: For two electrons separated by distance $r$, we have
$$F_g = \frac{Gm^2}{r^2}$$
and
$$F_e = \frac{1}{4\pi\epsilon_0}\frac{e^2}{r^2}$$
The ratio is
$$\frac{F_e}{F_g} = \frac{e^2}{4\pi\epsilon_0 G m^2}$$
Now choose a unit system in which $4\pi\epsilon_0 G = 1$, yielding
$$\frac{F_e}{F_g} = \frac{e^2}{m^2}$$
So the constant Feynman is referring to is the charge-to-mass ratio of the electron. I think most physicists consider it an important fundamental constant. It was first measured by J.J. Thomson in 1897.
I'll have to let others comment on what the universe might look like if $\frac{e}{m}$ had a different value.
A: 
Does this fundamental constant of nature have a name?

Not generally, as far as I know (which is not much...).

And does it have some profound significance in physics?

Again, probably not. The reasoning being that especially these two forces are such incredibly different regimes (as witnessed by the $10^{42}$).
The issue is that in large bodies (planets...), charge mostly cancels out - even if a celestial body carries a net charge for whatever reason, it is still "mostly" a mix of protons and electrons, and most of them will cancel. On the other hand, we know of no mechanism that would cancel out mass; every object whatsoever that constitute said large body contributes to its overall mass and hence gravity. In other ways, we have no indication that anti-gravity exist, while "anti-charge" exists just fine and is the default state of every usual atom (i.e., positive and negative charges in the form of protons and electrons).
So in anything but the quantum realm, those two aspects do not occur meaningfully at the same time. If you have a celestial body, you have gravity, and can more or less ignore charge. If you have something which is small enough to carry a meaningful charge (in relationship to its mass), then you don't care about gravity (in relatioship to the charge of course, not in absolute terms).
There might or might not be more to it on a quantum level, but we don't know. That's what a grand unified theory would hopefully clear up for us.

Is it another one of those "fine-tuned " constants? Or would a slight change in it not matter much?

See above. Until we have the GUT and know how these things work, there is no way to even guess. Still, since the constant is distance-independent, charge on a quantum level still is so much more "forceful" than gravity... hard to imagine that changing the constant even by a few zeroes would matter in any way whatsoever.
A: What he means when he says it's a fundamental constant is just that you can measure it anywhere in the universe and you'll get (as far as we can tell) the same value anywhere. The ratio of the volume of the earth to the volume of a flea can only be measured here, and also depends on the flea.
If it has a name, I don't know it. Rodney Dunning's answer calls it the charge/mass ratio of the electron, but it's different from that, not only because it's squared and has a factor of $1/4π\epsilon_0G$ but also because the $m$ in it is gravitational mass instead of inertial mass. (As far as we can tell, gravitational and inertial mass are equal, but the experiments to measure them are different.)
There are 26 constants in current physical theories, give or take. Feynman's constant normally isn't taken to be one of those, but it could be. Saying there are 26 constants really just means that the parameter space of the theory is 26-dimensional. Listing 26 particular constants amounts to choosing a coordinate system for that space. You could use Feynman's constant as one of the coordinates, though it isn't a standard choice.
The largeness of this constant (or its reciprocal) is necessary for our existence. If it was close to 1, as you might naturally expect it to be, then the universe would have recollapsed under its own self-gravity long before enough time had elapsed for planets to form and biological evolution to happen on them. I don't know whether it's fine-tuned in the sense that a smaller change (say a couple of orders of magnitude) would preclude our existence. It's not actually a well-defined question as stated, because the effect of varying a parameter depends on what other parameters you're holding constant (or in other words, what the other 25 coordinates in your coordinate system are).
A: This constant is the ratio between the fine structure constant $\alpha$ and the gravitational coupling constant $\alpha_G$. It is denoted by $N$ in Martin Rees's Six Numbers, but he mostly calls it "the big number".
It has pretty important effects on what kind of structures are possible in the universe. Basically it sets the size hierarchy scale between things, affects stellar formation and lifetimes, and the size of planets. To have life somewhat like ours only a portion of the $\alpha,\alpha_G$ plane is possible, generally implying that the ratio has to be small.
A: Yes.  This ratio is called the Dirac number.  In particular, speculations by Dirac led him to suggest that the gravitational constant $G$ varies as $1/t$.
For additional discussion of this there’s this review available on arXiv:

Ray S, Mukhopadhyay U, Ghosh PP. Large number hypothesis: A review. arXiv preprint arXiv:0705.1836. May 13 2007.

There’s also a very nice chapter on dimensional analysis, dimensionless constants and such numerology in

Barrow, John, and Frank Tipler. "The cosmological anthropic principle." (1986).

The difficulty with this numerology is that, if you look hard enough, you can find pretty much anything you want in terms of numerical coincidences (especially if you allow multiplication or divisions by geometrical dimensionless factors like $4\pi$ or $4\pi^2$).  The key is in finding supporting well-grounded physics arguments to explain the ratio.
A: The choice of two electrons is one of many possibilities. I can think of infinitely many charged particles, and even more combinations of these. Restricting ourselves to elementary particles, I still have 3 quark and lepton generations and two charged bosons, so 66 fundamental constants. You have to add fundamental constants based on other interactions. Why only take the ratio to gravity? Probably more than 100 fundamental constants result.
