Yes, it roughly means something like "a tiny bit" or "minimally".
It's an odd case where dictionary definitions aren't very useful since almost all the examples given are in negative clauses ("not in the least unfriendly", "wasn’t the least bit shy") - it's quite hard to find any examples of its usage in a positive clause. I found a few examples in other literature (all emphasis added by me):
She looked out at the pale desert. The twin white moons were rising. Cool water ran softly about her toes. She began to tremble just the least bit. She wanted very much to sit quietly here, soundless, not moving until this thing occurred, this thing expected all day, this thing that could not occur but might. A drift of song brushed through her mind.
-- Ray Bradbury, The Martian Chronicles
Major Monkey thanked him again. And then he said:
"I'd like to get acquainted with all the neighbors—such as they are. And I would suggest that you give a party and invite me and a lot of people to come to it, so I can meet them."
Old Mr. Crow bit his tongue. It struck him that Major Monkey was just the least bit too forward.
-- Arthur Scott Bailey, The Tale of Major Monkey
"But the rest's a necessity: not that I love new paint or am pining for electric light, but for reasons which I will just breathe in your private ear, Bunny. You must not try to take them too seriously; but the fact is, there is just the least bit of a twitter against me in this rookery of an Albany. It must have been started by that tame old bird, Policeman Mackenzie; it isn't very bad as yet, but it needn't be that to reach my ears. Well, it was open to me either to clear out altogether, and so confirm whatever happened to be in the air, or to go off for a time, under some arrangement which would give the authorities ample excuse for overhauling every inch of my rooms. Which would you have done, Bunny?"
[...]
And I left him without further parley, reading his impatience in his face. Everything, to be sure, seemed clear enough without that fuller discussion which I loved and Raffles hated. Yet I thought we might at least have dined together, and in my heart I felt just the least bit hurt, until it occurred to me as I drove to count the notes in my cigarette case. Resentment was impossible after that. The sum ran well into three figures, and it was plain that Raffles meant me to have a good time in his absence.
-- E.W. Hornung, "The Chest of Silver"
Perhaps from these examples you can understand better the meaning of "the least bit" in a positive context: it basically means "a little bit, a tiny bit, the least possible". It can also be used as an ironic understatement to mean "very much", as in the Major Monkey example above, but that doesn't seem to be the case in the example that you've quoted in your question.
