If you decide in advance that you're not going to want a particular mount or minion for a particular character, you can just avoid adding that mount/minion to your mount or minion list. You get them initially as items, and then "use" the item to add them to your list. Before doing so, the item version can be deleted (or stored away on your retainer in case you change your mind later). That way Mount Roulette and Minion Roulette will never see them.

Once they're on your mount list or minion list, though, there's no way to delete them. All you can do then is avoid the roulette and choose your mount or minion specifically, avoiding the ones you don't want that character to use.