Yes, they can refuse payment arrangements, that is part of their "firm demand for the balance in full" (collections training 101). I seriously doubt, however, that were she to send in a check/MO, that they would refuse to cash it. If an account is in collections, then would it not already be on her credit? They are allowed to only report accurate information on her credit report, unless she can negotiate something better.
What can she do about it?
-Try to negotiate with the CA. Start out lower than she can really pay, but I'd advise to never set up automatic check withdrawl, and never get a PDL to pay a bill. As a collector, I hated snail mail, but as a debtor, it's the only way I'll pay.
-Ensure her rights under the fdcpa and billing laws are not being violated (for example, threatening arrest, calling too often, incorrect amounts, etc...).
-If she's upset about the collector being rude, I'd advise your mom to be always be calm on the phone, and drag on the call as long as she possibly can. Time is money, so if the collector is being rude, but staying within the law, that is probably the best way she can get even with the person on the other end of the phone. In my office, no one gave a rat's butt about a collector being rude, I don't see that changing for any other call center, either.
Someone else should be along soon, and I'm sure they can offer more advice on negotiations. My experience was collecting a certain type of debt that was not negotiable--PIF was just about the only way to clear up the matter.