I'm sure someone will be along with more specific information but here's the little that I know. This is based on my cousin's football scholarship, not basketball, but I would imagine they're similar.
High school career was everything. My cousin played football practically since birth, and was pretty much already penciled in as "future star quarterback" when he hit 9th grade. He was moved up from JV to varsity pretty quick his freshman year and was a starter by sophomore year. I honestly don't think there was a time the recruiters weren't after him. He was just one of those "born to play" people.
The moving thing didn't come up for him, so I can't say for certain, but my impression is that it would depend on your character's rep. If he's already "known," like my cousin was, I doubt it would matter as long as he quickly becomes a rising star at the new school. If he's on the bubble, though, not really a known entity, then moving out of state could cause him to start over at the bottom, so to speak.
The new school may actually be the biggest factor. Does the coach there know him? Will he be a major part of the team right away? Or by moving, is he relegating himself to bench duty while he tries to make a name? His recruiter will matter too...does he have one? Is it someone who really believes in him?