On a slow melody, one form of variation is to take the chords which you (or an accompaniest) would fit to the tune, then play them as arpeggios, but taking key points from the melody, and making sure you include enough of the original tune to remind the audience that it is a variation.
Some tunes lend themselves to making variations. Planxty Irwin is one of my favourites for "messing about with".
Other "tricks" include stretching the rhytm, adding "rolls" of the notes above and below the melody note, adding "runs" to a target note, from a lower or higher note in the tune. The kind of "extras" you add will usually depend on what the instrument allows. Decorations on a banjo are very different from the rolls and trills which suit a tin whistle.
With many variations the important thing can be to keep enough of the tune to make it still recognisable.