As The Reid Report notes, Conyers refused to hold impeachment hearings against George W. Bush while he was the Chairman of the House Judiciary Committee, but suddenly sees it fit to sue Obama for acting legally.
In terms of policy, this is not complicated. The President has significant powers to conduct foreign policy under the framework of treaties the United States is a signatory to. Both the UN charter and NATO are such treaties, under which the President is acting. The action in Libya was sanctioned by the UN Security Council, and after 10 days or so of lead action, the US handed the lead over to NATO, falling back to a support role for NATO. The United States has obligations under both the UN and the NATO alliance. The President is clearly authorized to act, as both treaties are duly recognized as law in the United States under the Constitution.
Another thing is also clear as a matter of law and policy: Congress has the power to stop even legal use of force by the President through its control of the purse strings. Even when the President is perfectly within the law in using military force, Congress can simply say, "No, we don't like it; we're cutting off the funds." Members of Congress, Conyers included, are free to pursue that option, and the way the news media is drumming out the "bipartisan" nature of this group suing the President, why are Conyers and his friends such chickens? Why try to get the courts to do their dirty work - which, of course, courts will decline to do as they usually do not intervene in internal disputes between the executive and the legislative branches of government.
Come to think of it, the House actually already spoke on this. In May, the House voted 416-5 to bar President Obama from using any funds in the Defense appropriations bill to put ground forces in Libya. Note: they did not vote to cut off funding but rather to restrict the use of ground troops, which President Obama is not doing and has made clear that he has no intention of doing. If Congress wants to interject in this, this is their only way.
So this lawsuit is for one purpose and one purpose alone: pure show. Whoever said politics was show-business for ugly people is turning out to be right, even if the 'ugly' objective generally only applies to the motives, rather than physical appearances, of our politicians. You have Speaker of the House John Boehner expressing all sorts of "concerns" about Libya, an operation in concert with NATO and the international community that saved thousands of Libyan lives and enabled the country's own revel forces to fight back against their oppressive regime.
Frankly, if Congress has something to say about this further than they already have in the defense appropriations process, they have to say it in their power of the purse. Either grow a set and try to block funding for it, or shut up about it and stop making an ass out of yourselves and showing the Libyan people that you care more about your theatrics than their life. Put a damn cork in it.