A couple of exercises I find helpful:
Take a familiar song and, keeping the melody notes the same, change the rhythm so that the original song becomes (nearly) unrecognizable, and you've composed a "new" song.
See example below.
Same deal, but instead of changing the melody rhythm, change the accompaniment.
Combine the two
Change the rhythm and then the harmony, or vice versa, and you'll pretty well have a new song.
Keep in mind, these are just exercises. The goal isn't necessarily to create a song that will ever see the light of day; just a way to practice exploring combining melody, rhythm, and harmony.
Here's an example of "re-rhythmatization". I've taken a well-known song and "recomposed" it, keeping the pitches exactly the same, but changing the rhythm. IMO, the original song is nearly unrecognizable, but your mileage may differ.
T: "Mystery" Song
E8 z4 DCDE- | E z E2- E4 z4 EDz2 | D8 z8 | DEG2 G2ED- D8 |
C2DE z2 E2- E8 | EE2D- D z D2- D4 ED z2 | C16- | C16 |]