My cousin had a similar experience when, at Level 12, we pointed out that his horns were white while the rest of his Tauren was black. He immediately logged off, and met up with us an hour later with an all-black Tauren with the same name. Speaking of names, my wife leveled a Priest but decided at some point that she didn't like the name and rerolled. These were the days before paid character customization, of course.
I suppose we all have different reasons to reroll. Before the days of PvE-to-PvP transfers, a bunch of us decided to start from scratch and reroll on a PvP server because we couldn't progress on our original one. As painful as that was at the time (we all had a couple of Level 60 toons already), it turned out to be a great decision as we ended up being able to experience the endgame. Have you ever felt the need to reroll, or start from scratch? Not quite the same as leveling an alt, but to start fresh entirely. With paid server transfers and character customization, I'm certain the need is less urgent, but for those of you who started anew, what were your reasons?