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.
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.