I prefer a low sheen poly, SBF is also a good choice.
Mineral oil is more prone to getting water stains.
JMO

I would be worried about condensation that causes water to accumulate on a coaster as it is being used.  So definitely NOT mineral oil.  Poly would work but epoxy might be a better choice.  I wouldn't rule out pure Tung oil.

Re design - coasters are decorative and functional, but shouldn't be the source of excitement.  So glasses must sit securely with no tendency to rock from side to side.  I would make sure that the top of each coaster is dead flat, and also make the bottom slightly concave so that they sit on an outer rim without rocking.