if you have baseboard molding, you can remove a section of it on the
wall where you want to run the wire, drill a diagonal hole down and
inward through the base of the wall that will be covered by the
baseboard molding. Use that as your point of reference (you might want
to stick a piece of coathanger through your freshly drilled hole to
make it easier to spot from below). You'll then need to drill straight
upward from that vicinity and you should then hit the wall cavity. If
it's an exterior wall, chances are that it has insulation in it which
will make your job much harder. A wire fishing tape might be helpful
here if you can borrow one - push it up into the wall from below (a
helper above makes this much easier) and try to get the end of it to
come up through the hole in the wall where you want the thermostat to
go. Attach the wire to it and pull it back down. Sounds easy, but it
usually isn't. It'll come down to whether you want to fight with this
or patch drywall, and there's no right answer for it in my opinion.