1.) Trademarked name for a Milton Bradley boardgame.

2.) A plot device in the RuneQuest setting of Glorantha. A heroquest in this context is when a character glorifies the myths of his cult by acting out one of their dramas. Since this is fantasy, they often gain some relevant magical hero power, such as the ability to blind with a gaze for priestesses of Subere, or cheat Death, for priests of Yelm.

The hero power gained is commensurate with the challenge level of the drama. At one end of the scale, a heroquest drama can be played out in the purely mundane realm, which results in a good feeling for a few days. Kind of like going to church on Easter, or Christmas Eve.
More difficult is a quest against another cult, with an assault executed through the spirit plane, perhaps to disrupt their High Holy Day ceremonies. A victorious participant might gain an additional use of the divine magic Blind spell, or they may be awarded the service of a cult spirit.
At the "extreme danger" end of the scale, high-powered cultists can use magic to interact with the God Plane, perhaps even act out their god's dramas in God Time. Failure is disastrous in this level; unlike mundane adventures, Death in the God Plane is often permanent. Not only that, one god's failure in a myth is usually a hostile god's victory. Failing a highly-charged Hero Quest may mean doom to your whole community, by unleashing or failing to mitigate some hostile divine force. This is why Arkat said "no HeroQuesting without humility".