Republicans are invoking the Strom Thurmond rule as an excuse to deny President Obama his right to appoint a Supreme Court Justice. They aren't even going through the pretense of waiting until Obama nominates a new Justice before saying they intend to shoot him down.
Ted Cruz has vowed to filibuster any nominee President Obama picks so that the people can have a voice in the next election. Marco Rubio agrees. They both believe that a lame duck president in his last year in office should not be picking a new Justice. They want, instead, to delay the selection of the new Justice until after the next President is sworn in. As Senate Majority Leader Mitch McConnell writes
The American people should have a voice in the selection of their next Supreme Court Justice. Therefore, this vacancy should not be filled until we have a new President.
It is important to note that they have announced their intentions to shoot down ANYONE Obama nominates--whether he or she is qualified for the job or not. In doing so, they have declared their intentions to abrogate their responsibility. The Constitution says the President gets to choose a Supreme Court nominee and it delegates the Senate's role to giving its "Advice and Consent" on the nomination. For those who have reading comprehension problems, the phrase "Advice and Consent" means the Senate is supposed to ADVISE the President on the nominee and give their CONSENT unless they have specific reasons why they believe the nominee is not fit to do the job. Or, to put it in mathematical terms:
Advice and Consent = Advise and Consent
Advice and Consent DOES NOT EQUAL Block and Obstruct
Update: Obama should nominate Ted Cruz for the Supreme Court. Then he should sit back and watch to see if Ted Cruz will filibuster his own nomination. If Cruz does not filibuster himself then Obama should withdraw the nomination on the grounds that Cruz has proven himself untrustworthy and demonstrated that he can't be counted on to keep his promises.