Paul is the best traditional songwriter...and the most successful I can name more McCartney tunes than Lennon and Harrison combined, so he definitely has written the most memorable songs.
Lennon, however, wrote serious tunes where the meaning behind the song was more important than the song itself. That makes him a more relevent songwriter.
Harrison was underestimated since the early days of the Beatles, so he's surprised most of us when he succeeded with tunes like "Here Comes the Sun", "While My Guitar Gently Weeps" and "Something". While he wrote fewer songs compared to his bandmates, his material was stronger than Paul and John simply because he didn't release anything subpar. He had fewer albums, fewer singles, fewer duds.
Considering these factors I'd say McCartney is my favorite, but Lennon and Harrison are among my favorites in rock and roll. McCartney's versatile abilities prove he's easily accessible to many different types of music fans. He can be serious, but he can also be the goofiest card in the deck. Harrison and Lennon tended to craft the same image throughout their careers; album after album they kept the same attitude, and that grew very dull in my opinion.
McCartney gets my vote.