The idea of a long-term series with a strong opponent is a longshot, but I would vote for either UCLA or Texas, with the slight edge to UCLA. Why? First I eliminated teams who would likely not accept the offer due to current long-term series with strong OOC opponents (Stanford, USC, Florida, Florida State, and South Carolina). Then I eliminated teams with poor recruiting grounds (Arizona and Oregon). That leaves us with LSU, Texas, and UCLA.
These teams are all located in exceptionally fertile recruiting grounds. I think it would be harder to pull kids from Louisiana (since almost all the good talent there goes to SEC teams), so let's eliminate LSU. That leaves Texas and UCLA. Texas is a historical powerhouse, so that would be fun. However, if we played UCLA in Pasadena every other year, that would give us some familiarity with the Rose Bowl, which could be good for our team come the post-season. Also, UCLA is more similar to Michigan academically (two of the top three public research universities in the world--the other being Cal). It would also help with recruiting since players would be guaranteed to play in southern California two or three times in their career. Win-win-win.