I never had to pay anything to name my daycare. I was licensed and was even recommended by state to get accredited, I claimed my $$, etc. I had a name, tag line, and a logo. No payment, aside from my licensing fee so I could remain licensed.
Hmm...I suppose it's a state thing. Or maybe I am wrong and naming a daycare would not apply to naming a business..but I wouldn't see why not. At any rate check the state laws. In my state if you name a business you pay for the name or the paper work to process it..something like that.
If you decide to trademark it, or copywrite it, then yes it would cost money. Is that what you're thinking of?
In our state you have to register a business name with the state. Whether it is a daycare or what ever, they require registration with a fee. I would assume that when you got licensed they gave you the rules and regs?