In Texas, the mother of my husband's daughter was served child custody papers and instructed to get in touch with attorney in 21 days. What if she didn't make contact? Does she face any penalty?

More detail is needed. Was service confirmed? What is the nature of the custody case? Were the papers issued by a court?

If someone ignores a court summons, then its possible the other party will win the case by default. Its also possible a bench warrant might be issued for contempt of court. Or nothing might be done and second summons issued. Too few details to give a definitive answer.

You sure it was a summons, and not a certified letter? There is no penalty for ignoring your attorney, except he CAN file whatever his case is in court and they can issue a summons for her to appear before a judge.

If she's the mother of OP's husband's daughter, it appears that husband is suing her for custody. If so, husband's lawyers can have her defaulted if she fails to file something in the custody case within the time set in the summons ("court papers"). It would be unusual if she also was required to "get in touch with attorney".

If OP is asking about this in behalf of her husband, it looks like husband hasn't been fully advised by his lawyer about the process. He should call the lawyer and ask for clarification.