So far, Daniel Webster appears to be the all-time record holder, having argued 168 times. I didn't check this out but have no reason to doubt it, as lawyers didn't exactly fall all over themselves to argue before the Supremes in those days the way they do now, and he probably had the field more to himself.
Among more recent attorneys, the record holder appears to be Deputy Solicitor General Lawrence G. Wallace, who racked up 157 arguments over a 35-year career and just retired in 2002.
John W. Davis supposedly argued before the S Ct more than any other lawyer of the 1900's, or at least the first half, ending up on the wrong side of Brown in one of his last appearances. According to Findlaw, he was involved in 82 cases, but due to the way opinions used to be written, you can't really tell his oral appearances from those cases in which he was only on brief.
Numbers I got using Findlaw for other advocates known to be active are as follows:
Erwin Griswold - 79 oral arguments
Daniel Friedman - 77 oral arguments
Rex E. Lee - 53 oral arguments
Archibald Cox - 52 oral arguments
I'd be interested in knowing exactly how many cases Ted Olsen has argued, but again because of the way opinions are authored recently, am having a hard time figuring this out using findlaw.
There are some numbers for the supposedly "most active advocates" on the Oyez site, but they are WAY off. That site needs some serious work done on it.