It is simply the diminutive form of an obsolete word meaning "father," and is usually applied to all village priests.

The ushka is a common diminutive termination, and the root Bat is evidently the same as that which appears in the Latin pater.
Though I do not happen to know what Batushka's family name was, I can communicate two curious facts concerning it: he had not possessed it in his childhood, and it was not the same as his father's.
The reader whose intuitive powers have been preternaturally sharpened by a long course of sensation novels will probably leap to the conclusion that Batushka was a mysterious individual, very different from what he seemed--either the illegitimate son of some great personage, or a man of high birth who had committed some great sin, and who now sought oblivion and expiation in the humble duties of a parish priest.

Let me dispel at once all delusions of this kind.

Batushka was actually as well as legally the legitimate son of an ordinary parish priest, who was still living, about twenty miles off, and for many generations all his paternal and maternal ancestors, male and female, had belonged to the priestly caste.

He was thus a Levite of the purest water, and thoroughly Levitical in his character.