Arbitration eligibility and minor league options are two different things, although most of the time they coincide.
Arbitration eligibility is simple in HBD: Anybody with between 3 and 5 years of big league experience is arbitration eligible.
Minor league options are trickier. When a player is added to the 40-man for the first time, his options begin ticking. Each season that he is demoted from the big leagues or removed from the 40-man will result in the loss of an option year. If he doesn't get sent down in a season, he doesn't lose an option.