The courts will just dismiss this case for "lack of standing" as they did his father's lawsuit against Obama for violating the War Powers Act regarding Libya.
The Constitution provides a remedy for the Executive Branch violating laws, and it's not having the Legislative Branch go to the Judicial Branch. Congress should pass a veto-proof law clarifying its intention that universal wiretapping is against the law, and then if the Executive Branch persists, then start impeachment proceedings, where members of Congress act as judge and jury. Rand Paul's lawsuit is nothing but grandstanding -- similar to the conservative all-talk-no-results Republicans have been feeding their constituents for the past half-century, but this time it's libertarian all-talk-no-results. And unconstitutional to boot.
(Congress could conceivably start impeachment proceedings now without first passing clarifying legislation, but impeachment is a card that realistically can be played only once every couple of decades, so you want to make sure. If you don't have the votes for legislation, you sure aren't going to have them for impeachment. (You can also substitute "ethics and political will" for "votes".))