I am not a lawyer, and this is not legal advice. This also applies only to the USA, as I don't know how it works elsewhere.
However, I have successfully mounted a pro se defense against a spurious collections suit. So what I'm saying here is personal anecdote only.
If they're using this threat (and I have no doubt they are, having dealt with collectors before), then they are acting in a 100% illegal manner. Per the Fair Debt Collection Practices Act, if they don't have a court judgement against you, there is NOTHING they can legally do. They may eventually garnish wages, but NOT without a court order allowing that. And that only comes after a civil suit that's found in favor of the collectors. However, there is no way that anyone can be put in jail for failure to pay a debt (unless it's child support), and to even imply that is against the law.
She needs to document and, if at all possible, record each and every call she gets from a collector. Save everything they mail you as well. If they are threatening jail, that's a violation of the FDCPA. If they're threatening to garnish wages without a court order, that's a violation. If they're threatening ANYTHING they don't have the authority or ability to make good on, that's a violation. If they fail to issue the "mini-miranda" at the beginning of the call, that's a violation as well. By law, they are required to state something to the effect of "This communication is from a debt collector, and any information gained will be used for that purpose." They also have to verify that who they are speaking with is the person they intend to reach. They are not allowed to discuss any details whatsoever until they have verified this.
Each time they violate the FDCPA (and a call might have one or several violations), they can be liable for up to $1,000 in damages, payable to the party they are acting against, i.e. your mother. So if they call and all they say is something like "You owe $xxxx to us, and if you do not make arrangements to pay this debt, you will be arrested and placed in law enforcement custody.", they could be liable for up to $5,000 for that call alone.
1) They failed to identify themselves as debt collectors
2) They failed to verify that they have reached the correct individual
3) They discussed confidential information without verifying who they were speaking with
4) They threatened action that was not intended to or legally able to be taken
5) They falsely asserted that there would be jail time involved for failure to pay
Each and every one of those items listed is a seperate violation, and they would be liable for up to $1,000 per if it went in front of a judge. The legalities of call recording vary by state, but if they say the call may be recorded, that counts as their consent to recording. So if you're recording, that also counts as your consent, thus satisfying the two-party consent rule in some states. Also, some states like Illinois have loopholes that allow you to bypass the consent rules if you have reason to believe that the other party is or will be violating the law with the call. If you aren't able to record the call, still document the time and date of the call, along with what was said as close as you can possibly remember. If it comes down to a lawsuit, you may be able to subpoena their call records and recordings.
There is also the option of a Cease and Desist Letter, which tells them they are not to contact you at all except through legal counsel. This will stop the calls, but it could very well mean their next step is filing papers, so be aware.
Finally, write a validation letter to each and every collector a call or notice comes from, and send it via certified mail. Clearly state that the debt is being disputed, and ask them to provide full documented validation that the debt is indeed owed by her. In most cases, they will be required to provide the name of the original creditor, the original outstanding balance, as well as a signed instrument (such as a credit application or service contract). If they cannot provide these, they have no grounds to persist. Even if you know the debt is legitimate, this makes sure you're dealing with someone who is actually authorized to collect, and not some phishing expidition. If they persist without validation, guess what, that's another FDCPA violation right there. And even if they aren't found liable for damages, not having validation will usually lead to them dropping the case.
If you want to give the FDCPA a read for yourself, I highly recommend it. It's surprisingly to the point and easy to understand.Full FDCPA Text