Frequently Asked Questions - Green Card through Children
Yes you can. Having the same last name is not required.
You have several choices, including "No Last Name Used." If she wishes to use some other name, including adding a last name, now would be a good time to do it.
She can sponsor you for your green card upon turning 21.
1. You have to be a US citizen to sponsor her Green card. For Green Card holders as I recall the only family other than the children and wife they can sponsor is unmarried adult children over 21 but unmarried you can apply for their Green Card. Once you become US citizen your possible beneficiaries can be larger including your parents. There is little-known provision that is sometimes used, use it if it's necessary when let say your parents they don't have anybody in India to take care of them.
See clip from Attorney Rajiv S. Khanna's conference call video that addresses this question.
This is one thing very common in many cases. The issue of applying for a green card, when you are visiting on a tourist visa or even converting a tourist visa to H-1, L-1 and student visa is always a problem.