My mentor got laid off from a private research lab. Is there anything I can do to help him find a new job?
Your mentor most likely has his own list of references and if it comes to the point where he needs to have references from his grad students, then he is doing a very poor job of taking care of himself.

Offer to assist him in any way you can. If he needs a pair of extra hands to complete a side project that will get him hired where he wants to be hired, volunteer to be that extra pair of hands. Who knows, you might learn more from that experience.

Offer to be his eyes and ears for positions on your campus and have him say in the cover letter that he is applying on your recommendation.

If you want to offer help someone, talk to them and discuss what their need for help is and in what ways you can help meet that need. While you are assisting, leave them the lead in taking care of themselves. It is more economical of your time and energy and leads to a better outcome to help in the ways that you can help than to obsess about helping in ways that you can't - you are not in a position to give recommendations, so don't bang your head against the wall about giving one. Go for something doable instead.