Yes, they always do. I'm not sure of the true value of a player option but it's at least $200,000, because last season I was competing against a team offering a max contract and since my current budget only allowed me to offer $28.8 mil in the first year, I slapped a player option on him and he accepted my offer.
I prefer to offer mutual options since it provides an "out" if a guy's ratings take a nosedive when he gets old, but I'm not sure if those have any bearing on a player's preference for a contract. Player options (and no-trade clauses, too) definitely do, though.