Anyone can write a song if they choose to. The songs I've written are quite varied, but mostly I like to tell a story with my songs. So one important thing to have is a good story to tell. Once I have the story, I try to find an interesting angle to the story. Sometimes this means placing myself (the narrator) as a character who participated in the event. In other instances, I write the song from the viewpoint of a person who might have actually been there. And thirdly, the song must have a decent melody in order for it to be receive approval from the audiences where it is sung.

As George mentioned above, there are many more things that a song needs besides these three. I believe it is also important for a singer/songwriter to vary the type of songs that he writes. When performing your own material you should have a good variety of songs to sing so your audience doesn't get bored. So write ballads and love songs as well as humorous ditties. Have some of your songs written for accapella. Write songs in different keys, different chord progressions, different time signatures, etc. Don't write the same song over and over.

Best wishes,
Mike