I have come to love the Trail Name part of the Hiker World. I have a lot of fun with it, both in naming myself and talking to others about their names.
You can find a lot of stories and information on the Interwebs about trail names, but I haven't found yet any definitive origin of the trail name. In some of my readings I have seen that other groups (like fraternities and sororities) use nicknames and since a lot of the hikers nowadays seem to be college age or just out of college, maybe that was a natural extension of their nicknaming.
Regardless of the origin, it is a very popular part of hiking the trail and isn't just a Thru Hiker thing. Anyone who gets out there is able to take on a new moniker that they can take with themselves along the trail and beyond. Its popularity has grown along with the number of people who hike the trail.
To add on to what Larry Luxenberg said, I remember reading that Earl Shaffer called himself "The Crazy One" and later "The Original Crazy One". I think the meaning and origin of that one is evident as no one believed he was doing what he was doing and it took him a while to convince the ATC that he completed a Thru Hike.
Here are the reasons I have seen for Trail Names. It is a fun way to take on a new persona while hiking the trail. You can shed your "Real World" life and take on the Outdoorsy life along with a new name. You can name yourself (but don't tell the purists that) or a name can find you along the way. The one caution is, a name that finds you may not be as flattering as you want it to be, but if you are a good sport, you can go with it. One thing I have seen that a person who constantly vetoes suggestions, ends up with the trail name Veto. :)
For me, like I said, I like having fun with my trail name. The trail has named me and so have I. You get to think about a lot of useless things while out walking all day, so during this time I came up with my own Guidelines for Trail Names. I call them "Joe's HYOH Completely Non-Purist Guidelines for Trail Names". Only I follow these rules, but I like to explain them to others when I tell them I have several names (and one really long, strung together name) and I can interchange them or add to them at any time I feel like it.
1. I can have up to 14 trail names at one time.
2. I can add or delete to my trail names whenever the urge comes to me.
3. If I find a 15th trail name, I must select one to delete from my trail name list.
4. I can string my names together if I want and make one super long trail name.
5. I can create my own names.
6. I can accept suggested names, if they fit.
7. If the name fits, even if it is unflattering, I should always consider accepting it in the spirit of the name game.
8. Do your best to help others find their Trail Name(s).
9. Keep having fun, no matter what. As Model T says, "Find the Happy Side of Misery".
10. Everyone is entitled to a Trail Name. Even if they are out for a day hike. NO ONE has a monopoly on Trail Names.
And that is if for now.
El Jefe Maestro Obi Wan, the EarthToned Stiller Fan