Having done this several times, I prefer the clean start a close down and rebrand offers. While it is more than possible to continue under a given name and prosper, it does come with baggage, prior expectations and nostalgia.
My reasons for recreating Nisaba Syndicate as AWEX- were simple, AWEEX_ would be truly independent, unaffiliated with any previous partnerships and committed to a very different industrial model than the predecessor. The clear division between the two helped my members and I to disassociate ourselves from the previous ways of things and embrace the new.
So really it comes down to how open your members are for change and if you need to start a new page of history entirely. Adaptable members looking for a fresh view? Probably new corp. Adaptable members looking to retain nostalgia, but capable of a mental division between the old and new? You could keep the old name. If your members aren't adaptable, then both approaches are fairly doomed, but people seem to be fairly flexible once they've spent time in this game going up against real challenges.