I tried to research this myself and- well, got stuck in the mud. The word (or two closely spelled words that are conflated?) is given variously as “crossroads”, “debauchee” or “Mud Season”. Doubtless this makes more sense to a native Russian speaker but would someone sort it out for me please.
Rasput- = mud
-in = man
so Rasputin = man of the mud.
Common Russian naming convention: Lenin = man of the Lena (a river), Stalin = man of steel,…
This is the muddy season, and I suppose also related to crossroads.
Rasputin is his actual family name, and is generally supposed to be a toponymic, for a literal crossroad or fork.