Most states have rules regarding the naming of entities, and won't permit an entity to be filed if, for instance, the name is too similar to an existing entity or the name is misleading to the public. You can use your own name in the LLC if you wish; however, if you are starting a childcare business, the alternative would be to give it the same name as your business.
As my colleague mentioned, it is more important that the entity is properly formed, maintained and treated as a separate entity. As the purpose of setting up the LLC is to shield your personal assets, you want to avoid the potential for alter ego issues. You should set up a separate bank account and have an operating agreement. You should enter into any business contracts in the name of your LLC and not your name personally.
A good business attorney would be able to assist in the formation and advise you on the practicalities of having a single-member LLC.
One more thing to consider in this respect: if you use your own name for the LLC and then wish to use a different name for the daycare, you will have to register that different name as a "fictitious business name" or "d/b/a" with the clerk of the local superior court in each county in which you are doing business.
In this day and age, I normally do not suggest a client use their own name for the business. Be creative, consider the marketing aspect and find a business name that will grab potential clients. Your own name may feel good but may be less secure from a technology perspective.
DISCLAIMER This answer is for informational purposes only under the AVVO system, its terms and conditions. It is not intended as specific legal advice regarding your question. The answer could be different if all the facts were known. This answer does not establish an attorney client relationship. I am admitted only in the Commonweath of Massachusetts.