I activate Havengul Lich's ability targetting a Rattleclaw Mystic. I choose to play the Mystic as a Morph. Will my Lich be able to tap for mana this turn? More precisely, is Lich's trigger looks for the abilities of the creature card in the graveyard or for the abilities of the creature once it went on the stack?

Havengul Lich will gain the Mystic's mana ability until end of turn, since the Lich's delayed trigger looks for the abilities of the creature card as it existed in the graveyard.

Havengul Lich's ability targets a “creature card”, and then the delayed trigger it sets up gives the Lich the activated abilities of “that card”, and there's only one card it can possibly be referring to here: the one in the graveyard. If it was trying to refer to the spell you just cast, it would have said “that spell” instead.