The Republicans believe that suing Obama will excite and turn out their base voters, and they're betting that this benefit will be larger than any political blowback. Whether they're right in this political calculation or not remains to be seen. But what is undeniable is that, so far, this lawsuit is nothing short of laughable.
The GOP knows that crying, "Obama is cavalierly using his executive power to bypass Congress and legislate from the Oval Office!" will trigger a powerful public backlash and reinforce its usual charge against him of dictatorial abuse. It has played this card with maximum skill in its fierce fight to wrest back control of the Senate.