Set a timer for 20 minutes, and write down as many ideas as you can. Use whatever system works best for you, or no system at all. Write down the stupidest, most ridiculous song ideas first, just to get your mind moving. Try to free-associate your ideas. Allow each idea to be the inspiration for the next. You can build a great deal of momentum this way, and some of the connections you make could surprise you.
Listen to a song you like so much you wish you wrote it. Listen to the way the lyric interacts with the melody and note any structural similarities between them. Now re-write the lyrics using the original melody and chords. Note how the song is organized. Most songs are comprised of three verses, one chorus, and one bridge. If the original song has unique lyrics for the verse each time it is repeated, then this exercise will require three sets of verse lyrics as well. Try to emulate the way the original song incorporates imagery and metaphor.
Take the new set of lyrics and put them to a new melody. This is much easier if you can read and write staff notation. If you don't know how to write down a melody, use an audio recorder. Try to make this melody as different from the original as possible. Perhaps where the original melody ascends, yours could descend, or you could use different rhythms. Even though your lyrics were written with another melody in mind, the new melody will be better suited for them.
A good song has to have a sense of unity. When you listen to a melody, you should notice a good amount of repetition. A tasteful amount of repetition is essential to a good song. For example, write or sing a four-bar melodic phrase. Then, over the next four bars, repeat the contour and rhythm of that phrase while moving all the notes up a step (within the key). This is motivic expansion, and it is often used by songwriters whether they realize it or not. Just as one idea can inspire the next, or a lyric can inspire a melody and vice versa, any element of a song can inspire any other element.