Picking 'the speed of light in a vacuum' does seem to be fairly arbitrary, almost as if physicists had decided to write equations in terms of 'the top speed of a stock 1972 Dodge Viper', but as I understand it, light really does have a special place in the scheme of the universe. Newton's laws are an excellent description of the way the universe works at the scales humans work with, but with extremes of mass, energy or velocity Newton's laws break down.
Einstein's theories have a number of consequences that seem strange to our limited human experience; things like time running slower for things moving at an appreciable proportion of c, the mass of fast-moving objects approaching inifinity as their velocity approaches c, and time running lower in the vicinity of very large masses. Despite how strange these predictions sound, they seem to match our observations of the world around us - for example, one of Einstein's early triumphs was that he was able to accurately predict the orbit of Mercury (deep within the Sun's enormous gravitational field) when Newton's laws could never get it quite right.
As for 'why light specifically', well, it isn't exactly - it's the speed of all electromagnetic radiation. Of the four fundamental forces in the universe, half (strong and weak nuclear forces) only work on atomic scales, and gravity works by its own special rules, leaving electromagnetism as pretty fundamentally important.