There is no "rule", the name is an asset of the partnership and subject to joint use, no use, or whatever unilateral use you can work out. Goodwill is as ambiguous as to amount and may be zero.
The above is general legal and business analysis. It is not "legal advice" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.
This is not unusual. In a general partnership, when a partner decides to leave, the partnership is dissolved. Dissolving a partnership requires partners to equally split the debts and assets of the partnership. Partners who desire to remain in business after a partner leaves can establish buy-sell agreements when first forming the partnership. A buy-sell agreement allows the remaining partners to buy the ownership rights of the departing partner and this normally includes keeping the name.
This is my opinion and should not be construed as legal advise for your specific case as there are many more facts which you have not provided.
If the former partners cannot agree on who can continue to use the trade name then the first to re-use the name -- either by actually doing business under the name or adopting it for a new company that properly registers with the Secretary of State -- owns the name. A trade name, like a trademark, does not simply exist in the ether. It must be used to exist. Because partnership disputes are very fact specific, you need to speak with a trademark attorney to flesh out the details of your particular situation.
The above is general information ONLY and is not legal advice, does not form an attorney-client relationship, and should NOT be relied upon to take or refrain from taking any action. I am not your attorney. You should seek the advice of competent counsel before taking any action related to your inquiry.