A woman usually can choose to affix the last name of her husband to her name or replace her maiden last name with that of her husband’s. In India, the latter option is often preferred. But name change after marriage isn’t like a compulsory thing.
If your mother sure, and ready for a name change after marriage, then she need to first get a marriage certificate, which means she need to legally get her marriage registered. She will have to download a form from your state government’s website and submit it at Sub Registrar Office. She will have to submit an array of photocopies of documents like Aadhar Card, PAN card etc., whose list she can find on the government’s website. Please note that marriages in India are registered under the Hindu Marriage Act, 1955 or the Special Marriage Act, 1954. You can find more details about each on the state government’s website or the central government’s website.