Why doesn't the conservation of angular momentum modify mass instead of velocity? Background
Kepler's second law says that planets sweep equal areas in equal times in their orbits; the closer a planet is to its parent body, the faster it moves; the farther it is, the slower it moves. This is explained by the conservation of angular momentum, which is $m * v * r$. 
$m$ is the planet's mass
$v$ is the planet's orbital velocity
$r$ is its distance from the parent body
As $r$ decreases, $v$ must increase to maintain the momentum.
Question
Why is $v$ "chosen by the Universe" to be the value that changes in this phenomena and never $m$? I've only heard of mass being modified theoretically, when something is attempting to increase $v$ beyond the speed of light. If that's true, why does mass remain immutable until that point?
 A: The universe doesn't make such assumptions.  We do.  In the majority of the problems you will use conservation of angular momentum for, the nature of the problem dictates that the mass will remain the same.  Mass doesn't appear out of nowhere!  Mass is conserved.  Velocity is not.
There are plenty of physics problems which involve the sudden addition of mass (such as if someone throws an object at you and you catch it).  In these cases, mass can change rather than velocity.  Or both can change!
As for why mass is conserved, that's a philosophical question which stems beyond science.  That's where we have to start admitting that things like "conservation of mass" is a model of how reality works, not the definition of how reality works.  However, if we are willing to handwave that aside and say that "conservation of mass" is such a good model that we can treat it as though it is reality, then we're back to my first paragraph -- we assumed the mass didn't change, because the model says it wont.
