Start with the lyrics. Elton John has written some of the world's most recognizable melodies and has stated in interviews that he almost never starts with a melody, but writes his melodies to lyrics written by his partner, Bernie Taupin. Let the rhythm and sound of the words shape the contour of the melody. Unique combinations of words and phrases lend themselves well to equally unique melodies.
On the other hand, you can write great, effective melodies to very simple lyrics. The choruses to Paul McCartney's "Silly Love Songs" and Whitney Houston's "I Will Always Love You," written by Dolly Parton, contain very similar, simple lyrics, but make for memorable melodies and hit singles.
Listen to popular songs and those you like and determine what it is that makes them memorable. Use these melodies as a guide for your own. Take the first few notes of one melody and branch off and create your own. Try singing new melodies over the chords of existing songs, but never outright copy a melody. George Harrison was sued for subconsciously copying the melody of The Chiffons' "He's So Fine" in his song, "My Sweet Lord." Be sure not to copy a melody, but don't be too concerned about a melody sounding like a pre-existing one. There are only so many notes and combinations of chords possible, so similar-sounding melodies are an inevitability. Famous songwriters are even known for ripping off their own melodies. The verses of John Lennon–penned hits "All You Need is Love" and "Instant Karma" are strikingly simliar, as are Prince's "Manic Monday" and "1999."
Write as often as possible. Leave your mind open to the possibilities of new melodies, whether playing a chord progression on the guitar or standing in line at the supermarket. Allow the environment to soak in and influence melodies. The sound of a passing ambulance can trigger a melody just as well as the strum of a few chords.
Take note of all possible melodies. Carry a notebook or some sort of recording device at all times in order to preserve the melodies. If you can, write out or hum the intended chord progression with the melody, as you may remember the melody later, but not the exact context of it with a harmony. Focus on the ones that seem to pop up in your head more often, and work on crafting those melodies. If you can't remember it, its usually not catchy enough to keep focusing on.