J.D. Rock and roll history is filled with litigation involving the rights in distinctive names for musical groups. Since the name of a band can be a significant asset, often the single most important asset owned by the band, it is extremely wise to settle these questions before the band becomes popular. Once there is money thrown into the pot, the negotiations often become considerably harder and more costly.
Poverty often makes deal making easier.
Unless a group expressly agrees to the contrary, or absent some special circumstances I will go into in a bit, the members that become the original band, all of whom essentially join at the same time, are considered in law to be partners with the right to share equally in the profits of the partnership. This means that all members are deemed to be equal owners in all the assets of the partnership, the group name being one of those assets. And this applies no matter who actually came up with the name for the band since all partners blend their contributions to the general partnership asset pool. This means that upon the dissolution of the band, all members of that partnership have an equal right to use the band's name in the pursuit of their new recording careers. This is subject to a number of qualifications including that the use of the name must not be misleading to the potential consumer.