Let’s put aside standing for the moment. Why would the House of Representatives bother suing over the President’s action on immigration when a federal district court is likely (in my estimation) going to issue a preliminary injunction in the next two weeks. This case will be quickly appealed to the 5th Circuit, and very likely be decided by the Supreme Court before July. By the time the House takes a vote to authorize the suit, and prepares the complaint, the case will already be halfway to the Supreme Court! Disregarding any political advantages, strategically, this is a bad move because it distracts from the states, who (I argue) do have standing.
The House should sit this one out. They can file an amicus in the Circuit and Supreme Court, and move for divided argument time at the Supreme Court. They very well may get that. This gets them around the entire standing burden, and they can make their case.
Disclosure: I joined a brief in support of the states in Texas v. United States.