Im pretty sure its just the dara from yamatagi maDARA. The kids just shortened it to be something easier to say and to be a cuter name.I can't believe it took me till now to realize that Dara-san name is taken from Kankandara, the japanese internet based legend.
With her being an enshrined Yokai and all that worship having a direct tangible effect on her, the misogiya siblings are probably making her both stronger and more stable. Meaning evil snake Yamatagi has less and less of a chance at ever regaining control.
Not just is the evil snake basically doomed to never take over Dara anymore (she accepted the new name and everything, she's basically reborn as a guardian deity), he is also being changed by them and being turned into Dara-san's subordinate as a guardian of the children.