Minor league options are one of the most confusing items in baseball. Here's a quick rundown of how they work in HBD:
All players start with 3 at the beginning of their career. The first time a player is added to a club's 40-man roster, his option clock starts ticking. Any subsequent season in which the player accrues 20 or more days on optional assignment (i.e. days on a minor league club from the start of the regular season until the team's final game including playoffs), an option year will be burned. If he's on the big league club for the season or on optional assignment for less than 20 days during the season, he doesn't burn an option.
During a season in which a player has options, he may be demoted/recalled at will. Once his options are gone, he must go through waivers before he may be demoted or removed from the 40-man roster. Once a player has 6 years of big league experience, options are meaningless because he can opt for free-agency rather than accept a trip to the minors.