Like most of you, I'm learning on the fly how these appointments work, but let me inject some facts in this discussion, - some pros and cons:
•Supreme Court appointments are rarely actually rejected. The last was Reagan's appointment of Robert Bork in 1987, which was rejected by a Democratic Senate in Reagan's last year of office, so a similar, but reversed situation to what we have now. Reagan's second appointee was forced to withdraw because of a scandal and finally Justice Kennedy was approved unanimously.
•Even lame duck presidents eventually get a nominee in, even if it is not their first nominee.
•An exception can be the so-called "Thurmond Rule" which according to Wikipedia "is an informal and somewhat amorphous rule in the United States Senate regarding confirmations of judicial nominees. The rule has been described as being that "no lifetime judicial appointments would move in the last six months or so of a lame-duck presidency." This is not really a rule, it is more like a political reality.
•The record for longest time period to go through the process is 125 days, although this one could set a new record. Obama has about 342 remaining in office. I think our best hope is that Obama nominates someone they can find grounds to reject and they take their time doing it. Then the next nominee starts pretty close to that six month window. Then they could drag their feet until the clock runs out and not actually have to reject the second nominee.
•More bad news: Obama could appoint a nominee during Senate recess and they have three big ones coming up: Summer, Fall and Winter. That appointment would only last for two years, though, not a lifetime. However, a Democratic Senate blocked Bush from making recess appointments using "pro forma" sessions, where one Senator stays behind on recess, opens the Senate each day to an empty chamber and they closes it ten minutes later. The Senate is technically still in session, so the president can't make a recess appointment, but nothing gets done.