Is it possible for my mother to stay in the US while waiting for her green card?

Assuming you are a US citizen and that you are filing for your mother, she may not have to return to Haiti for green card interview and will instead adjust her status [green card] while she is here in the US. This is based on the assumptions that (a) your mother is now physically in the US and (b) she came to the US with a visa and was inspected and admitted by an immigration officer at the port of entry. If so, you should file the I-130 petition together with the I-485 [Green card ] at the same time along with the I-765 work permit. She would receive the work permit within 90 days of filing the paperwork and you should expect her green card interview no later than 8 months. To ensure you are doing the proper things, consult with an attorney to avoid possible implications further down the road .

If you are a US citizen and your mother is already in the US as a visitor and has been here for at least 90 days, then you can petition for her and she will be able to remain in the US until she gets her green card.

You need to make an appointment with a competent immigration attorney to fully discuss the risks and the benefits of such a choice. Your mother is not legally allowed to remain in the US past the I-94 date regardless of the visa's length. She may be eligible to change status to another category which would then keep her in status. It may be possible for her to remain in the US and still adjust status but there are a lot of risks there and it is not allowed under law. For this reason, please make that appointment.

Probably if she came here legally and you are a citizen.

As long as you are a US citizen, then mom can stay and complete the process in the US.