The show was originally going to only run for six weeks. By the end, the question of who this guy was would be answered. When it suddenly became popular (due to the Daleks) they kept it running, and dropped the notion of explaining the character.

Jerry and MEM are correct. It's a long-standing traditonal joke that, in at least every incarnation, at some point, someone is going to recite the "Doctor? Doctor who?" line. The latest being Tennant's first episode (The Christmas Invasion) when Rose says that this new guy is The Doctor, and Jackie (her mom) replies "What do you mean that's The Doctor? Doctor who?!"

There have other various instances where it's been hinted that "Who" may be a shortened translation of his last name, much like when he met the female Time Lord Romanaveradnalunda. The Doctor explained that he wasn't going to call her that everytime, and said he would shorten it to either "Romana" or "Fred". She liked Fred, but he stuck her with Romana.

In his 3rd incarnation, when he was exiled to Earth, the license plate on his car read "Who1" (or "Who3", I can't remember). Other times, conversations have gone like this:

"I'm The Doctor."

"The Doctor? Doctor who?"

"Yes, quite right..."

Add variation.

And, finally, the credits for probably close to fifteen years of the series listed the character as "Doctor Who".

So, there's plenty of precedent for why it's called Doctor Who. It may be his name. It may be a title. It may just be the name of the show. But, in the end, the only name he'll ever offer is simply just "The Doctor".

Besides, calling him "Doctor Who" is kind of gay...