He does have a last name, but we're never told (I believe).
Once on Nickelodeon they had a specially made clip in the advert brake:
Arnold was filming himself and he was talking about a few things, like, "This is a shirt, not a skirt" (referring to his shirt that hungs down around his legs). Then he goes "And yes, I do have a last name, it's....". It cuts out with fuzz (as he was filming on a video camera)
I don't think they would do this is they had ever given it away previously.... I haven't seen the nwer episodes, but I surpose they're following on with this theme, and we probably won't ever know.... Maybe the cartoonists never even gave him a second name??