Your family member already has been sued (I assume this means "served"), so the first order of business is to file an answer. Above all else, file the answer within the time allowed by law, or all other options will become moot. It is very important that all defenses be included in the answer because the defendant may be prevented from raising them at a later stage. Take the time to draft a complete answer, which in this case would include the affirmative defense of the statute of limitations. You might want to consult an attorney at this time.
Going on to the issue of proof of last payment, this is what trials are for. This is the fun part of my work: the litigation of juicy claims. First the answer, then a brief round of discovery requesting a copy of all documentation of payments and a copy of the contract in the hope that there is an attorneys fees provision. This forces the collection agency to put its cards on the table or get out of the game. Then, after the case has been constructed, an active round of settlement discussions. That failing, then trial, which should not be a problem if the discovery has been done properly.
A debt collector is not a judge. You (your relative) must "tell it to the judge." that involves filing a response to the lawsuit, propounding discovery requests in accordance with the Code of Civil Procedure, compelling compliance with discovery requests, and going to trial. The SOL is an affirmative defense, which means that the defendant has the burden of proof. The case will be decided by the evidence produced in court, and by the testimony of witnesses. If you win the lawsuit, the credit card agreement may contain an attorney's fees clause. If so, the other side can be ordered to pay your attorney's fees.
I strongly urge you to consult a lawyer for a more specific evaluation of your case.
The statute of limitations is an affirmative defense that must be asserted properly or else it is waived. What your family member should do is contact a consumer rights attorney like me ASAP. It is also possible to file a lawsuit against them for misrepresenting the nature and character of the alleged debt. I would love discuss this with you in more detail, as I actively pursue harassment lawsuits against debt collectors all over SoCal. Please check out my website www.californiadebtharassmentattorney.co