General mission:
* Misyon (formal)
* Gawain (task, assignment)
Mission as in a religious mission:
* Misyon (formal)
* Pangangaral (preaching)
* Ebanghelyo (gospel)
Mission as in a military mission:
* Misyon (formal)
* Operasyon (operation)
Mission as in a goal or purpose:
* Layunin (goal, objective)
* Hangarin (aspiration)
The best choice will depend on the specific context you are using the word in.