If you are doing babysitting yourself, why are you looking for a company name? You need YOUR OWN name known, referrals from families you babysit for, recommendations of the "My babysitter, Jane, is terrific - very reliable, great with the kids, and always coaxes them to bed on time so I don't have cranky, tired kids in the morning!"
Why spend money on getting a name, logo, trademark, company organization when you babysit?
If you were starting an agency with licensed, bonded, reference-checked sitters, nannies, etc, then you need to do all the company formation, logo, trademark, business licensing, etc. Plan to spend $2000-5000 to start up. Of course, you'll also need insurance, and that will cost big time for such an operation.