Wrong Apple ID keeps popping up asking for my password

It sounds like there is an old Apple ID stuck in your phone. If it's yours then log it in then log it out and then login your current one. If you don't know whose it is then perhaps the previous owner didn't fully logout and that could be problematic.

Can you provide anymore detail on your purchase of the device?

Do you have any apps on your iPad from that account ? All apps (and anything else that is downloaded from the stores) are tied to the account that downloaded them - delete the apps linked to that account and buy and download them on your current account.

Or if it's the previous version of the same account try : What to do after you change your Apple ID email address or password - Apple Support

My problem is similar, except I don't know what app or what tune is attached to these various IDs.

Some of these items were offered to me from another user of Apple devices. I don't remember what and I'm not in touch with these users anymore.

I have deleted all apps that could be associated with the old ID's and I'm still getting prompted for passwords for the wrong itunes ID.

Check your account's purchase histories (assuming that you haven't bought items on multiple accounts) : See your purchase history in the iTunes Store on a Mac or PC - Apple Support

Go through your purchase history and delete anything that's not on it

You are 100% sure that you've deleted all of that account's contend from the iPad ? If you are try a soft-reset/reboot