Babs’s name day is December 4, the feast of Barbara the Great Martyr.
Churches that follow the Old (Julian) Calendar keep the same feast on December 17 of the civil calendar. See everything commemorated on December 4.
The saint behind the name

Barbara the Great Martyr
December 4, Martyr, Βαρβάρα η Μεγαλομάρτυς
St. Barbara the Great Martyr lived in the late third century. According to tradition, she was the daughter of a wealthy pagan, Dioskoros, who confined her in a tower; coming to faith in Christ, she confessed Him openly, and after cruel tortures she was put to death by her own father, upon whom judgment swiftly fell.
Also celebrating on December 4
These names share the feast, many of them other forms of the same name:
Varvara, Varoula, Barb, Barbie, Barbara
What to say on a name day
The Greek wish is Χρόνια Πολλά (Chronia Polla), “many years.” A call or a short message on the day is what people remember. See name day wishes you can send and what a name day is.