What is so special about speed of light in vacuum?

Special Relativity is based on the invariance of a quantity called the proper time, $\tau$, which is the time measured by a freely moving (i.e. not accelerated) observer. The proper time is defined by:

$$ c^2d\tau^2 = c^2dt^2 - dx^2 - dy^2 - dz^2 $$

This is similar to Pythagoras' theorem as learned by generations of schoolchildren, except that it includes time (converted to a distance by multiplying by $c$) and it has a mixture of plus and minus signs. The mixture of signs is responsible for all the weird effects like time dilation and length contraction, and because there is a mixture of signs the value of $d\tau^2$ can be positive, negative or zero.

If $d\tau^2$ is less than zero then $d\tau$ must be imaginary, and therefore unphysical. A quick bit of maths will show you that $d\tau^2$ can only be negative if you travel faster than light, and therefore that $c$ is the fastest speed anything in the universe can travel.

So $c$ is special because it determines a fundamental symmetry of the universe.

Footnote:

I've said $c$ is special while Kostya has said the opposite, but actually we are both right.

Kostya is right that there is nothing special about the speed 299,792,458 m/s (though if you change it by much you'll change physics enough that we may not be here :-). However the speed at which light travels is very special because anything travelling at this speed follows a null geodesic, i.e. $d\tau^2 = 0$. This is the sense in I mean that $c$ is special.

Nothing. From Nature's perspective speed of light is entirely artificial number.

Imagine that you've discovered an alien culture that measured horizontal length $\ell$ and height $h$ in different units. They live on a planet with very strong gravitational force, and for them it is very difficult to rotate stuff in vertical plane. Such kind of rotations are really unnatural and counter-intuitive for alien-layman. While alien-physicists discovered them and have introduced a special transition coefficient $\alpha$ that transformed one dimension into another, allowing aliens to understand that both of these quantities are just projections of a more general thing called "distance": $$ d^2 = \alpha^2 h^2 + \ell^2 $$

Then imagine that there is an alien-physics.stackexchange site and someone asked there "What is so special about that $\alpha$?" And the answer is, again, "nothing". Nothing is special about $\alpha$ -- these aliens are just used to special conditions.

Same thing applies to homo sapience -- we are just used to very low speeds, which makes us think that time and space are completely unrelated and cannot be "rotated" into each other. While non-alien-physicists discovered that this is not the case, introducing a transition coefficient $c$...

Einstein in 1905 derived $c$ from the Maxwell's equations. Exact title of the paper in which special relativity was published was On the Electrodynamics of Moving Bodies. He essentially just resolved the problem of two electrons moving relative to each other. Moving electrons create a changing magnetic field and at the same time they are accelerated by Lorentz force and electrical force originating from the field.

In an inertial frame of reference associated with any electron there is no magnetic force ($q\mathbf{v}\times\mathbf{B}$, $\mathbf{v}$ being $\mathbf{0}$), only electric field acts on the electron. The solution is: if we change our frame of reference to a frame of reference which is moving at constant speed there is a transfer between magnetic and electric fields. There inherently is a constant in the transformation which units are metres per second. It was called "an Einstein's constant".

Einstein looked at speed of electromagnetic waves calculated using well known at the time magnetic and electric constants (the same formula as for Einstein's constant, by the way), and noted that resemblance with observed speed of light is... intriguing. It was just a suggestion of light being an electromagnetic wave.

So:

• Einstein's constant
• speed of light

two separate things. But since light is an electromagnetic wave, it's speed in vacuum is equal to Einstein's constant $c$.

Today commonly we call it just "speed of light" for convenience, but in many contexts it could be called "Einstein's constant".

$$c = \sqrt{\frac{1}{\epsilon_0\mu_0}}$$