What does the number 22 stands for in the F-22 fighter jet name?

It doesn't stand for anything, it is just the number assigned since the U.S. reset the numbering system for aircraft (which had been in use since before the Korean War) in 1962 by Defense Secretary Robert McNamara. Numbering for all military aircraft restarted at -1 after that (however the lowest number used for a fighter was the F-4, but the next bomber was the B-1).

Not every aircraft assigned a number went into production and use (e,g, the prototype Airforce F-17 program was terminated when the F-16 was approved and the fighter was redesigned to make the Navy/Marines F-18). Also some numbers were skipped without ever having an airplane assigned that number (e.g. F-1, F-2, F-3, F-19) and the F-117 was inexplicably assigned an arbitrary out of sequence number that appears to be from the old numbering system.