Mom put house in hers and my oldest brother's name several years ago. He doesn't want the responsibility. What to do?

Another vote for consulting with a Real Estate attorney.

It's unclear to me whether or not any conveyance actually occurred through the "giving" of the house to one brother.  How did the deed read before she made the decision, and how does it read now?  My point is whether or not your mother still holds title or your brother does.

This is the key issue before legal action can take place.    An attorney would ask the same questions; he/she will need to know specifically who holds title.

My experience is still very new, however, it does appear that you need legal advice.
If your mom doesn't have a trust, I would highly recommend you help her to get one in place. With her appointed executor, POA as well, you can facilitate her wishes.
Best of luck!

Depending on your state that will depend on how you transfer the title and how it’s held. I am in California, so if I hold my home with someone else as Joint Tenants, it reverts to the opposite person immediately upon my death. If we were to hold the home as Tenants in Common, then whoever is my established beneficiary or heir would receive my part/interest of the home upon my death. Brother may sign off his interest in the property But I believe to add somebody to it would require your mothers approval being that she is the other owner (state specific so research your state). Please do not take this as legal advice it is just my experience on how titles are held and transferred in my state. The mortgage or anything owing belongs to the person who took that loan and has no bearing on holding ownership BUT must be satisfied prior to the home being sold altogether regardless whose name is on the Title. You can call a local Title or Escrow company or stop in and ask their exact protocols, since they collect the notary fees and county recording fees to file with Recorders office I’d expect a Title Officer might give ya 10 minutes of their time just to assist. Once things are established between you’all you can contact same Title Officer and have them start preparing the documents for you, schedule an appointment, and pay for the notary and the recording fees so that it’s filed correctly. It’s quite painless. I live in something called a wet state we use an escrow officer who acts as our liaison and deals with the Title Company. Either way a quick phone call to one or the other and certainly someone can direct you which division to speak to regarding the matter. Hope this helps.

Contact an attorney who specializes in Real Estate Law. You want everything to be done the right way so there are no issues down the road with things like Probate, etc. This isn’t a do-it-yourself project.