Yes, you can claim Child Care Expenses you paid to your mother. You will need to list her SSN and the amount you paid her on your return, so she might need to report the income on her tax return as self-employment income if she babysits in her home and you paid her more than $600. But if you claim her as a dependent on your return (since you said you give her money), you can't claim child care expenses you paid to her.
Whether this amount of income would disqualify her from SSI (or reduce her benefits) is a question she would need to discuss with Social Security.
However, if your mother lives with you, she would be considered a household employee, but grandparents are exempt from the 'nanny tax' so you are not required to withhold or pay social security or household employee taxes, you would just put the gross pay in box 1 and box 16 for your state on her W-2.
She may still need to report the income on her return if she is required to file a return.
Here's info about Tax Filing Requirements.
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