Obama is renowned for his playing 11-dimensional chess. I predict a good move would be to nominate Olympia Snowe for Supreme Court.
She's not nearly as liberal as I'd like, but if the nominee is not a Republican, the nomination will be held up by the Party of No.
I fear a situation where the party of No-Government-Unless-WE-Get-To-Run-It blocking the Supreme Court nomination, and paralyzing the Supreme Court as well as the Senate (Breyer, Kennedy, Ginsburg, and Sotomeyer would deadlock with Roberts, Alito, Thomas, and Stevens).
If the GOP let Snowe's nomination through, Maine voters could remedy the clusterfuck that Martha Coakley created in the Senate.
For GOP members to filibuster her, they would be violating Reagan's 11th commandment (Don't speak ill of another Republican), and preventing another Republican from joining the SCOTUS.
As far as what kind of justice Snowe would be, she's socially liberal and her views that are more conservative (favoring foreign military endeavors and cutting taxes) are usually not before the Supreme Court.
Again, would I be most happy with Obama nominating Snowe to SCOTUS? No. But I think it would be a fiendishly clever move, and might allow him to get way more done.