grammaticality - Why is it "on *the* one hand"?

The definite article is used to signal that one is talking about specific items, not items in general.

For example, this exchange at the reception desk of the hotel.

Guest: I'd like a room.

Clerk: Will you be staying the weekend, sir?

Guest: No, just the one night.

The guest could have said "No, just one night," but adding the definite article makes it more emphatic and specific at the same time. This applies to "on the one hand" constructions as well.