Problem one was that the faucet was kinda gross looking, encrusted with minerals along the seam line, cloudy where it should have been transparent, and the light-up LEDs had long ago bit the dust.
Problem two was that the drain had some ugly rust spots and wasn't actually holding water in the bowl.
And the mystery half-problem was some evidence of water leaking into the underside of the countertop around the drain.
As I disassembled things, it became quite apparent that the half problem was actually a whole one. The not-very-good-at-its-two-jobs pop up drain turned out to be one for an overflow sink, not a non-overflow sink as this one is.
That meant that the water draining from the sink also drained out the overflow and into the particle board making up the counter top. Bit of advice: NEVER USE AN OVERFLOW DRAIN ON A NON-OVERFLOW SINK (or vice versa). Unless you want to turn your vanity into a really gross petri dish.
Anyway, a little bit of cleaning, figuring, and plumbers putty later and things look quite a bit better with a new faucet and correct non-overflow style drain.
Not sure if I'm going to do anything about the vanity top, the bit that's gone soft is kinda hidden away under the top layer of solid surface material and with time it'll dry out and become inert once more, so I'm thinking I'll just ignore it.