Well, most musical traditions that I'mm familiar with have two kinds of variations. The kind that involves playing around with the melody is one, where the musician embellishes the tune in some way.Decorations, perhaps: elaborating a slow moving tune with fast notes in between the main ones; changing the rhythm, eg with syncopation; etc etc.
    The other kind discards the melody, and plays around with the harmonic structure of the tune: ie you play something that doesnt sound like the tune, but which fits with chords which the accompanists are playing behind the tune(which tends to mean,in practise, that your variation will probably be a harmony to the tune). In non-harmonic music this kind of variation may involve playing in the same mode(or scale, or raga or whatever) as the tune. rather than with its chord sequence.
Early jazz lead players used to play the first kind of variation(round the tune) but the habit soon spread of palying variations on the harmonies. Theme and variation piecses(whether folky or classical) commonly use both techniques(or a mixture). In the 17th and 18th centuries in England the term for this was "divisions", rather than variations: a good name, because the technique involved dividing the notes of the tune into faster notes to produce the decorative effect.
    Well, there's a bit of classification. Not much help in telling you how to actually do it, but without being able to display the dots or play examples, I dont feel I can add anything helpful like that.