Why would you want to name a computer?
The same reason one would want to name a ship, I suppose. It's a tool, yet its owners frequently (in the case of people like me) have a very close relationship with it, and perceive it as having human qualities or tendencies. Corporations or Universities with large farms frequently name the computers to have an easy, eloquent way of referring to them, one which is more descriptive than just using numbers.

What kind of name should I pick?
The name of your computer should be a reflection of your own personality and mannerisms, and you perceive your computer. For example, all my boxen are named after moons. europa, ganymede, deimos, callisto. When I was choosing their names, I was in the midst of reading Arthur C. Clarke's 2010: odyssey two, so I was somewhat inspired by that. Some people like to name their computers after characters or concepts from a favorite book or anime. Some like abstract concepts, such as mathematical terms. Some prefer names from Greek or Norse mythology. Basically, it's up to you, totally. Be warned, however, that you simply cannot change the name once it sticks. Nothing is stopping you, it's just bad luck. Like renaming a ship after it's already been christened as something else. Also, naming computers after real-life people is very bad luck. I'm not sure why, but it just is.

How do I go about implementing the name?
Under Unix, there is a command which will basically set the name of the computer. Become root, and in the terminal do:

# hostname (yourname).(yourdomain.dom)
You may omit the yourdomain.dom if you wish, it doesn't really make much of a difference. Under MacOS, you may name the computer in the File Sharing control panel (called Sharing Setup under OS 7.x and below). Under Windows...well, if you use Windows chances are you don't care for your computer enough to give it a name. There is, to my knowledge, no facility to name your computer under Windows (cue myriad of uppity Windows users correcting me...)
Additionally, if you happen to have DNS control over the domain the machine is in, you may want to add an "A" or "CNAME" entry for it. Consult your BIND reference of choice on how to configure the nameserver.

And again, choose carefully. Computers resent being given a bad name...