I might be able to guide you a little bit Sensei since I'm getting plenty of practice building my Jimdo website. ^_^
From this url:
http://anime-beta.com/?reqp=1&reqr=
Y'all clearly have 3 options:
1. Get the login, passcode, and renew it. Have the owner initiate the domain transfer ownership to the current person nominated as owner.
2. Go ahead and Buy the domain from GoDaddy. (Not sure how that would work out forum-wise with all the stuff we've got on there now.)
3. Call Customer Service and Initiate Domain Transfer of Ownership and Renew GoDaddy as the hosting site. OR... go Host hunting once domain is secured and transferred to new owner.
Fees will likely be involved no matter how you look at it. Currently my hosting is free at Jimdo, for example. I can choose to upgrade to a higher tier membership, but it's going to cost a fee every month.
This can be a possible scenario for you that the owner may have to pay a monthly fee to keep Beta up and running if Free Web Hosting is not secured.
I vaguely remember mentioning that y'all MIGHT have to go Web Host Shopping or suggested we move it, even though I am not an admin. Either way we look at this, I agree Sensei that the previous owner should have delegated this task to the current nominated owner.