FROM THE SCHOOL OF SCHOENBERG: Adolph Weiss' Nine Ways of Varying a Musical Motive: ~ 1) changing the intervals or notes & holding the rhythms, 2) changing the rhythm & using the same tones or intervals, 3) simultaneous combination of both these methods, 4) inversion, 5) elongation, 6) contraction, 7) elision (of one or more notes), 8) interpolation (of one or more notes), 9) the crab form (motus cancrizans, repeating the motive backwards).Both for myself, and with students, I like to start with Weiss' list and invent or discover some other methods of variation. For example, there is the Duchamp method, in which the individual notes (both pitches or durations) are lifted from a measure, phrase, section etc. and then put back in again in the score by chance operations. Then there is the Clarence Barlow style of elision (related to Cage's erasures) in which the rests are substituted for notes (in his Çoğluotobüsişletmesi, Barlow automated the selection of "shoveouts" and "shoveinagains" in order to thin out or thicken a texture). I like another method of working with sequences or rows of pitches, which retains the Schoenbergian idea of waiting to begin again until all of the other items in the sequence or row have been stated, but does so not by a formal operation on the whole set but rather so that, in a 12-tone sequence for example, a note will repeat every 12 tones on average, but only as an average, sometimes repeating every 10 or 11 or 13 or 14 tones, creating small and familiar neighborhoods of pitches, but without every having an exact repetition. (To be honest, I'm most fond of this when used in diatonic or near-diatonic sequences). There are also plenty of possibilities in using contours (i.e. melodic shape, without a metric) as the basic of variations (this was an area pioneered by the composer David Feldman, and later a hot subject in music theory).
*I don't have a copy handy, but I believe that Marion Bauer's Twentieth Century Music (1933) had a similar list with an attribution to Weiss; Weiss -- Schoenberg's first American student and, for a time, John Cage's teacher -- is a figure about whose activities, as teacher or as composer, we know too little.