As the comment says: You pay mom and what she does with the money is up to her once she reports it on an income tax return.
So if you pay her a large amount and SHE pays for the lodging, you may consider the entire amount as day care cost for either the FSA or day care credit. But you may not pay for the lodging directly and call it day care expense.
But, from a practical standpoint, the math doesn't workout. There's a $3000 limit ($6000 for 2 or more kids) for claiming the day care credit and $5000 for FSA. Grandma would have to declare the entire amount as income and pay both income tax and self employment tax. She would not be allowed to deduct the lodging cost, as an expense.
