Strong merit/Strong Math dept for student who lives/breathes math

I have a friend who’s going to Rice next year as an undergrad math major. His dad is a mathematician who got his PhD from UCLA.

All these are great for undergrad math. I don’t think they offer the significant amount of merit that OP needs though. UMD , for example, has only a few substantial merit awards and they are very competitive.

Most REU opportunities require students to be a US permanent resident or US citizen.

Yes REU limits are another issue; however some opportunities are open to non citizens. Once he’s admitted somewhere, it’ll be time to see what connections exist for summer programs.
UIUC and UT have almost no merit that I know of. Texas Tech does but I don’t think it’s very good for math. UTDallas also does, who knows whether it’s good for Math?