This is a complicated situation, unfortunately it may cost more to engage proper legal help than the $1000-$1500 of income tax you are fighting over.
First, the most important thing is to never let an IRS letter go unanswered. If they send you a letter that says "You forgot to report this $5000 income and so we are assessing extra tax" you usually have about 30 days to send a reply. In this case, you would reply with details as to why the debt does not belong to you and the 1099 was issued in error. That gets your case in front of a human. If you don't reply by the deadline, then you generally are deemed to have accepted the results of the letter and agreed to the tax and you lose your rights to appeal. If that has already happened, you can still try and send a letter but you may have lost the argument before it ever started.
Then, in your IRS letter, you will explain the circumstances and include copies of any relevant documents. For example, you will want a copy of the cardholder agreement (if you can get one) that says that authorized users are not responsible. If you can find a similar statement in your state consumer law, you might include a copy of the law. Include copies of any correspondence you sent to the lender telling them you were not responsible.
Additionally, if the lender issued a 1099-C to you or your mother, but is continuing collections efforts, they are in violation of IRS regulations. The 1099-C is the final stop, it is a declaration that you owe, that collection efforts have stopped, and therefore what you owe becomes taxable income. They are breaking the law if they try to collect after issuing a 1099-C. You could include that in your IRS letter if they are still trying to collect.
Separately, there are other things you should be doing. You need to check all your credit reports, and file objections if this is listed as your debt. You can also contact your state banking regulators and file a complaint against the lender.
There is one case where you might actually be liable for this debt. Your mother's estate is supposed to pay all lawful debts before distributing her property. If this debt was not paid when she died, but she had more than $5000 in the bank or property worth more than $5000, then the debt probably should have been paid before distributing the property. If you received her assets (bank balance, stocks, car etc) without the estate first paying the debt, then you may in fact be liable, although it would usually take a court hearing to determine that fact--just being a credit card user would probably not be enough. But it's an issue that needs to be considered.
Lastly, because a 1099-C is a final determination, you don't owe the bank any money even if you were lawfully responsible for the debt. The bank can't collect, and you don't owe the money. Paying the income tax on the money closes the issue forever, if a 1099-C is issued.
