Rhys is one of my favorite boy's names of all time. (Which says something, because my favorite names change about every week).
I'd pair
Rhys with something classic and strong:
Rhys AlexanderRhys CorbinRhys AddisonRhys OliverRhys EmmanuelRhys MontgomeryRhys WilsonRhys HartleyRhys WalterRhys WilliamRhys MaxwellRhys MortimerRhys GideonRhys AdrienRhys Wallace
I think it would sound best with something that starts with an 'A' or a 'W'.
My personal favs are
Rhys Alexander and
Rhys Wilson.