How do I find out who trying to sue me?

In most cases, the case will be filed with the Court when the paperwork is served. If this is the case, you can contact the court clerk and look up the caption of the case using your mother's name.

You are correct in assuming the process server was hired by the attorney filing the complaint against your mother. You should begin by contacting the court clerk in the county court and the district court in the county in which your mother lives, that is probably where the claim against your mother is filed.

However, there is more to the story. Many attorneys who file lawsuits do not file the complaint until after they serve the defendant for the very reason that they don't necessarily want you to know who is suing.
In my experience, people being visited by process servers have already been contacted repeatedly by the plaintiff. So I would talk with mom about who she's been getting mail from.
If your mother has a lot of debts and problems that she can't get out from under, she may want to think about bankruptcy.
Lots of myths are floating around about bankruptcy. Bankruptcy is easier than it has ever been to file for the vast majority of people. Lots of things can be done in a bankruptcy, get rid of judgments, liens, second mortgages on houses, etc. Usually if you have lots of debt collection problems your credit report will actually go up after a bankruptcy.
Many law offices will consult with her for free about bankruptcy or handling her debts.
By the way, some attorneys sell "unbundled" or "limited" legal services where you pay for just what you need and can afford. For example, some attorneys will prepare letters for you to sign, or sell the paperwork for the court filings; then you can proceed on your own, but knowing that your paperwork is correct and having a road map as to how to proceed. Or who will attend a hearing for a flat fee even if they are not handling the whole case.
Good Luck, jim
Jim@NeighborhoodLawOffice.com