My mother is elderly, in poor health, and only income is social security. She has a large credit card debt (one card) and the company wants minimum payments of almost half her income. She has ask them to reduce tha payment, but they refuse. What can be done?

Go down to which property and income is protected from debt collection
Unless it is a government debt or a school loan this pretty much applies

http://www.lawhelp.org/documents/1386210200EN.pdf?stateabbrev=/WA/

Social Security Benefit Checks are Exempt from Garnishment, but…. (http://www.thebklawyer.com/thebkblog/2007/03/07/social-security-benefit-checks-are-exempt-from-garnishment-but/)

Well first have her cut the card up and not charge things she can't afford. Perhaps you can work extra and help her pay for the debts of things she actually bought and paid for.

And in the end, work with a credit counseling company to help arrange a payment plan you can afford