There have been a few LDS kids that have chosen major programs over the years... Seems like a lot of them were in the mid 90's. Jordan Bachynski (Arizona State grad in 2014) would have been a really nice addition the last couple years:

Jabari Parker, Duke
Chris Burgess, Duke (later Utah)
Matt Christensen, Duke
Travis Knight, UConn
Mark Pope, Kentucky (Washington his first 2 years)
Mark Madsen, Stanford (Stanford was a top 10-program during the Madsen/Jacobsen days)
Nick Robinson, Stanford

There have been some really talented less-active members over the years too. Maybe if our program was in a P5 Conference, some of the talented, less-actives would consider BYU. All four of these guys played in the NBA, and two of them were even all-stars:

Casey Jacobsen, Stanford
Kiki Vandeweghe, UCLA
Scott Pollard, Kansas
Tom Chambers, Utah