It's always good to have at least three farm names you like.....perhaps variations of the same name.....written down. Then call each registry and ask them to check if any of them are already registered.
When we did it, (15+ years ago), we lucked out and I quickly registered our farm name:
MINIATURE VENTURES with the prefix "Mini V" right over the phone with a credit card!
I think it costs in the $50 range, but it's registered forever.
Also, if you are serious about using your farm name as a business or as a breeding "hobby" you should contact your state to register your name with them as well.
MA