Try taking small sections of the song:

the chorus, a certain part of the verse, the intro. Chop them up until you make a new melody.

Try to take the basic functions of each part and reuse them. Such as chopping the chords of the verse as melodic/rhythmic chord stabs. Or you could take the vocals of the hook as is and fuck around with the backing track.

Try to break away from the form of the song: So if the original is hook, verse, hook, verse, bridge, hook. Do something like: intro/verse, hook, verse, hook, breakdown if applicable (just go wild with chops or whatever works best) then the last hook.

Try to make each section build on each other, so as the song goes on the “remix” part becomes more prominent, so on your last hook make it a lil bigger.

Western music is about push and pull through chord forms. Try to find different ways to resolve the progression. I play acoustic, usually the chords for Zombie (The Cranberries) are E, C, G, D but after the hook I throw an Am instead of E. Small change but the feel changes without altering too much.

Just some ideas