In my opinion, I believe the best thing to do is call the credit card company, explain that you need to close the account because the person is now deceased. My experience has been 1) "We are sorry for your loss, however, there is a balance owed" - my response - "not my responsiblity" or 2) "We are sorry for your loss, thank you for calling, would you mind mailing us a death certificate", my response - "certainly, do you need an original death certificate or will a copy of it be okay?"
I have had one company say "there is a balance owed, when will you be paying the balance" - my response - "No, I'm not responsible but I can mail you a copy of the death certificate". And then the company said "Well, we will go after the estate" - my response - "no problem, after the final expenses, there will be nothing left, so feel free to file against the estate". (Even if there is monies in the estate, it is just a tactic some tacky companies will try - there is no logical reason for them to "waste the money" to file against an Estate when they don't know the size of the Estate). I have had no one file against the Estate - credit card company or any other business. Family relations - yes. Businesses - no.