He actually went so far as to claim:
Men who become Pope change their names precisely to show that what they taught before, which may be erring, is of no account to their pontificate.Of course, my correspondent was just trying weasel out of the evidence that was stacked against him, but it did make me wonder: when did the name changing begin?
The obvious answer of "Simon's name was changed to Peter" isn't correct. First of all, "Peter" was Simon's surname (see the discussion here). Second, many of the early bishops of Rome (and alleged bishops of Rome) did not take on a pseudonym.
I'd love to have a more definitive answer, but EWTN reports this:
which seems like a reasonable explanation (source). The practice seems to have stuck, although I'm not aware of any canon law that absolutely requires a name change.
Papal Names - Most of the early Popes kept their own names upon election. However, when the Roman priest Mercury was elected in 533 he took the name John II, so the Church would not have a Pope named after a pagan god. Thus began the practice of taking a new name which today is taken for granted.
The last pope not to change his name was Marcellus II (crowned in 1555). Ironically, his name is the name of a pagan god (Mars, like his name sake and like pope Mark). It's also worth noting that while he would have been the first recorded bishop of Rome with the name "Mercury," the official list of popes also includes among his predecessors not only those named for the major god Mars, but also those named for some of the lesser gods: Dionysis, Anterus, and Zephyrinus. (source)
John II did not get the trend to catch on immediately. The next several popes maintained their birth name, though John III (originally Catelinus) followed suit (he may have changed his name before becoming pope). (source) Indeed, the next few name changes were to be called "John" (Octavian became John XII in 955 and Pietro Canepanova became John XIV in 983) (source). However, by 1503, when Julius II retained his birth name he was disrupting a 494 year tradition spanning 72 popes (source).
So - what is the real reason that popes change their names? It's a tradition. If you like it, thank Johns II, III, XII, and XIV for paving the way - but don't make the facile assumption that they do it for theological reasons, or that this is a tradition that comes down from the apostolic era.