@Felipe Ocampo ,
To answer your question, I would like to make a distinction between contract (lease agreement), and application.
The purpose of signing a contract (lease agreement) is to hold that person liable if things go sour between you and the tenant. If her son is going to live with her in that house, you can certainly get all the adults who are going to live in the rental house to sign the contract.
The other big piece of the puzzle before letting a tenant move-in is who is going to pay the monthly rent. You usually use the application to run the credit check and background check. She told you up front that she is retired, and I assume she has a fixed income. I appreciate honesty... she is not stupid. If her son is not going to live with her, does her son willing to be her guarantor? Are you prepared to take this risk?
You are the landlord, and it is your house. So, you make the rules. Some landlords don't want to deal with dog owners, and most landlords don't want to deal with people with bad credit / low-income tenants (except for Michelle and Bruce Fisher, https://www.biggerpockets.com/renewsblog/2014/07/1...)
Of course, if you are a new landlord, it is not recommended to deal with chasing your tenant for rents every month or eviction process.
Good luck