The variation I have heard the most, although it's not one of my favorite things, has been the substitution of the tune of "House of the Rising Sun". My wife has sung Amazing Grace to the tune "The Water is Wide" a few times; I takes just a little tweaking, as:
Amazing grace, how sweet the sound,
that saved a drifting soul like me;
I once was lost, but now am found;
I once was blind, but now I see.

We added one home-made verse when doing that:
Though grace may grow familiar now,
Like any song, sung one old way,
Lord, touch my heart, lest I forget,
Thy grace is new, each breaking day.