NATOLI-LEGAL, LLC
There is no easy or straight forward answer to this question.
You need to consider the trademark (brand) as an asset that has value (or perhaps not). If it has value and the business entity is dissolving you need make sure that it is considered among other assets and whether the entity has creditors. Assuming it does not, the best way to handle this is to enter into a written agreement that assigns all rights to the mark over to the member/partner that wants it. If this is a registered trademark then an assignment will need to be drafted and a recordation will need to be done at the USPTO as well.
If you cannot agree on a resolution and money value this is the kind of matter that may end in court and of course it gets very expensive. If the real market value of the trademark far exceeds what the one member/partner is willing to pay for it, then it should be sold and the proceeds split in accordance with ownership. A court may force that sale case depending. A court may also force one member to accept terms of the other, or even rule that neither member can use the mark until it is satisfactorily “abandoned,” which may take several years. So working this out in a reasonable fashion is the best outcome.
Before you make any commitments or take any action on your own, get some solid legal advice so you fully understand your rights and your best options.