It looks like a general Chassidic custom.
The Minchas Elazar writes that it is a tradition from the Ba'al Shem Tov. He explains the mishna that says that one stays is "comparable to a slave who mixes wine for his master and he spills it in his face." in a novel way.
He says that there are two ways to explain "he spills in his face", is it the master spilling into the slave's face, or the slave spilling into the master's face.
If the clause means "the slave spills it in into the master's face", it is saying that sitting in the sukka during rain is a chutzpa and is wrong.
However, if the clause means "the master spills it into the slave's face", it means that one who stays could still be considered a "midas chassidus".
Moreover, the Minchas Elazar says that one should enjoy the mitzva so much that the rain shouldn't bother him. And even though he may be a "hedyot", Jews are holy and willing to be hedyotos for the sake of a mitzvah.
Chernobyl, Skver and Koidanov also have such a custom, as well as Bobov (unless it's a public meal).