One way to measure Rep. Ron Paul’s ascendance as a political player is to compare the cold shoulder he got from rival Republican presidential candidate John McCain in 2008 with the cozier embrace he has received from 2012 presumptive nominee Mitt Romney.

When Mr. Romney announced Monday that he supports auditing the Federal Reserve, it underscored the odd but powerful influence of the maverick congressman from Texas, whose years-long push for a Fed audit will now appear in the party platform.

From overseas adventurism to deficit spending to a distrust of government institutions, the gospel of Paul has revived long-simmering debates that were muffled during the eight-year tenure of President George W. Bush — and, at least on some of those issues, Mr. Paul’s side appears to have gained the upper hand.