As executor of a trust, if the person you are executor of defaults on their mortgage loan, is the executor liable?

The word "Executor" is used only with a Probate Estate. The person who controls the investments and distributions from a trust is called the Trustee. The Trustee of a trust owes a duty to the beneficiaries to act in good faith, with knowledge of the facts, and using reasonable judgment. The fact that you are even asking this question suggests that your husband may not know enough about the particulars of the trust. For example, what his powers are, the purpose of the Trust, whether it's a grantor trust, if he needs to get a mortgage to protect the trust on the home equity loan, and more. I strongly suggest your husband get legal counsel to discuss these matters. Perhaps the attorney who drafted the trust in the first place. Perhaps an entirely different person who does not have a potential conflict of interest between his mother and him as the Trustee. Hope that helps.