Visitor visa refused. Could a family visa be accepted?

The UK Family visa would not be the right fit, as it is intended for those who wish to reside with a family member (spouse/partner, fiancé, child, parent) and for more than 6 months. In this case, your girlfriend intends to stay for a short time with an aunt and visit you (neither of whom was mentioned in the earlier application): the Standard Visitor visa is the appropriate category.

Absent a copy of the refusal notice and the section(s) under which her application was refused, generally she would need to address the cited issues. From what you say, her mother’s sponsorship was insufficient. While having a sponsor is certainly acceptable, it does not reduce or alleviate the applicant’s burden to show that they qualify in their own right. Indeed, having a sponsor may raise the bar. A new sponsor(s) won’t change that and the information in the Sponsorship Issues portion of this TSE response should help her understand that item.

Having relatives and friends in the UK is a perfectly valid reason to visit. Her failure to reveal that is now a problem she has to confront in reapplying (and she absolutely should include it; not doing so may make things worse). The ECO may have questions and concerns about her reliability, as it was a deliberate omission (everyone says otherwise, when caught), but it's better to anticipate and address them, rather than risk a second refusal for deception, a very bad thing.