Public deposit rules 58a

In the your matter client father is depositer & Tax payer, so his mother  can't sign on 15G or  15H

(If the depositor is not liable to pay income tax and the interest to be
paid/credited in a financial year does not exceed the maximum amount,
which is not chargeable to income tax, the depositor may submit a
declaration in Form No 15G so that income tax is not deducted at source.
Senior citizens (65 years and above) may submit a declaration in Form No
15H. Form 15G/15H should only be furnished by the depositors before the
prescribed date by the issuer.)

Dear Friend,

In your case, since your father has expired and your mother became the owner of fixed deposit with the bank. Your mother can file Form 15G for lower or non deduction of TDS.

Thanks.