Great question!
Info. I prefer:
Your horse: Size, breed, sex, temperament, how he keeps, any special needs.
You: What you’re looking for facility-wise, service-wise, and want to do.
The BO needs to know if you’ll be a good fit; if most of her business is people taking lessons, she’ll prioritize someone like that over someone who just wants a stall. If she goes to shows, she’ll probably prefer a like-minded client. You never know, though–I know of full-board places who also happily board retirees. The main point is to determine if she has what you want.
Do your mental homework first: What ideally would you like, what would you compromise on (indoor vs. turnout, for example) and what would be an absolute deal-breaker. Also, what you can afford to pay. Don’t forget to ask whether board is all-inclusive or whether there are add-ons for things like holding for farrier & vet, grooming, laundry, etc. Expect that lessons will be extra unless you’re in a program for training board. If that will be the case, be clear on what that includes.
Tell the potential BO all useful information about how your horse has kept in the past, why that did or didn’t work for him or you, and what you’re seeking to replace it. If you’re heavily into supplements or alternative therapies, let her know. Are you willing to use her vet and farrier, or are you strongly motivated to keep your own? Why you’re moving is also useful information. Expect BO to check references; vet and farrier at least if not past trainer or barn if moving in from out of the area. EVERYONE wants a harmonious deal that makes all parties happy–not to import a problem or try to jam a square peg in a round hole.
If you’re going to ride every day, only on weekends, have the trainer ride, bring in your own trainer, or expect to bring your kids or guests to the barn, these you need to mention also. Many barns welcome them; some don’t. Ideally, you want to feel totally welcome, comfortable, and appreciated as a client in your new barn.
If you have some big drama in your past, keep a lid on it. If you give TMI about a divorce, money problem, bat-crazy SO or big fight at your last barn, don’t be surprised if the place you’re interviewing “just filled the stall.” If you sound like more than they want to deal with, you probably are! Make sure to clean up any outstanding vet or farrier bills before you leave your old place, too, so they can give you a good reference.
Good Luck!