Listen to as much music as possible, both old and new. Limiting your listening to one genre of music or only modern music will seriously limit your songwriting. For example, even if you play heavy metal music, listen to folk, classical, soul and other styles to get more ideas. Take cues from hit songwriters you admire and attempt to emulate their work in your own.
Work on songs everyday, if possible. Even if you don't have any great ideas, set aside time to sit there and think of melodies and lyrics.
Learn music theory. If possible, take courses in music theory and/or read books on music theory. Knowing which notes work with which chords will improve your songwriting and help you figure out where to advance your song.
Approach each songwriting session differently. For example, if you tend to write songs starting with the melody, try starting with a set of lyrics. If you usually start with the lyrics, try starting with a simple chord progression.
Say something with your lyrics. Good lyrics tell a story and avoid cliche while still resounding with the listener. Rewrite your lyrics until they say something in a way that's never been done before.
Write catchy melodies. The element of a song people usually remember the most, the part they walk away humming, is the melody. Make your melodies hummable. If your melody is too static, try different series of notes to change it up.
Incorporate tried and true song structures. Verse-chorus-verse-chorus-bridge-chorus is a commonly used song structure, for example. Writing songs using standard structures will give your song a solid foundation.
Play your songs for your friends and at open mike nights. Ask for criticism and take note of any valid points about weak spots in your songs. Co-write with a peer if possible.
Go back and listen to all of the songs you have written. Analyze the ways in which you have improved, as well as your problem areas.
Craft your songs based on your own critique, as well as that of your peers. Continue to write everyday to keep your songwriting muscles toned.