1 Named after the university's Facebook
A Facebook is collection of names and photos of people distributed in American universities. It's a directory, usually printed, to help students know the names of each other. Because Facebook had a similar functionality when it started, Zuckerberg chose that name.
2 It was inspired by Facemash
Zuckerberg was inspired to create Facebook after creating his site Facemash, which used to compare college students to each other based on attractiveness. This event could also have been attributed to choosing such a name. See was Facebook really Zuckerberg's idea.
3 Facebook started on a smaller scale
When Mark Zuckerberg started Facebook his ultimate goal was to connect Harvard students together. He used the name TheFacebook because he wanted to replace the original Facebook with his site. See how Zuckerberg came with the idea of Facebook.
4 The name was changed to Facebook
Facebook was initially called TheFacebook, but when Sean Parker and Mark Zuckerberg met Parker suggested changing the name to Facebook for the sake of simplicity and Zuckerberg liked the idea. See how Parker and Zuckerberg met.
5 Students wanted a universal Facebook
According to WIkipedia, Zuckerberg said when he first launched TheFacebook: 'Everyone’s been talking a lot about a universal face book within Harvard. I think it’s kind of silly that it would take the university a couple of years to get around to doing it, as I can do it better than they can'.