I purchased turbo tax deluxe on line and prepared my tax return. How can I use the program again to prepare my mother's taxes?

When you use Online TurboTax, you get one return per fee.  Each return needs its own account and user ID.   If you use the same account and user ID for a second return, you overwrite the first one and lose it forever.

https://ttlc.intuit.com/community/using-turbotax/help/how-do-i-start-another-return-in-turbotax-online/00/25596