How is it meaningless? It forces Zero to go on record as vetoing it and thus accepting final responsibility for the fiscal profligacy. It's a point that can be made in a fifteen second ad.
The House will pass the “rejection” of the debt increase. The Senate will not vote at all, the time period will expire, and Obama will have to do nothing.
Meanwhile, because we botched the argument so badly, nobody except us cares that the debt limit is being increased, so nobody will be held “to blame” for increasing the debt limit.
Which makes sense, because by and large the voters don’t seem to be holding anybody “to blame” for actually increasing the debt either.