Geoff pretty much covered most of what I do to make variations on a tune.

The key ingredient is to figure out what notes are essential in the tune to make it recognizable, and messing around with the other stuff. Or leaving them out entirely. For example:

1. in The Rights of Man, play just the first note of each of the series of triplets that open the tune. (You can almost always do that with a riff of triplets)
   
2. listen to the intro of Loreena McKennit's version of Bonny Portmore--there's a pipe thing that is just a very selective part of the tune.
   

When you expect to hear something, your ears fill in the blanks.