Can my son's grandmother give him her car if she is on Medicare?

It should just be considered a gift. But depending on the value of the car and what state you are in it it could be different. It would be safe to check with an elder law attorney in your area. Here is one I know of locally californiaelderabuselawyer.

Gramma can give anything she owns to anybody she wants to. Depending on the value, the gift may have tax consequences, and also impact Gramma's eligibility for Medicaid.

I'm sure Medicare doesn't have a thing to do with gifting. You mean Medicaid.

I wanted to add to my previous answer, that you need to watch what you give away or sell if she is nearing the time when she might need Medicaid. Anything you do 5 years before could effect her eligibility when she applies for Medicaid.

I worked for an elderlaw attorney for 11 years. No! it is a gift. Don't give away any assets or do anything without talking to an Certified elderlaw attorney you could lose your medicaid eligibility. http://www.nelf.org/

As long as the car is worth less than $10,000 he should be able to avoid a gift tax.

Medicare does not have any gift restrictions. If she is on Medicaid, or will be within the look-back period, the fair market value of the car could be treated as a transfer penalty.