I was wondering what was the meaning of the tile Usagi drop: literally it means "Bunny drop", but how is it related to the story? Is there some reference to somethings or is there some pun involved?
As senshin has mentioned in his comment, the author has never commented on the origin of the name. Thus, there are only fans' interpretation...
Usagi Drop → Usagi's drop → rabbit's (falling) teardrop
The first link explains more in detail that in the last volume of the manga,
Daikichi confessed to Rin why he took her over.
Rin: Daikichi... you haven't met me before, but why you...
Daikichi: Your face... your crying face when you parted with grandpa... somehow...
The next panel shows Rin's crying face. Thus, Usagi = Rin.
The conclusion is, the title is about Daikichi's determination to take Rin over when he saw her crying face.
Also, the anime's OP show a teardrop just before the title appears.
The second link explains it in short, that the title is a wordplay.
"Usagi" means lonely, and "Drop" means teardrop.
Combining the words, the title means Rin, a lonely crybaby.
Further reading: in Japan, there's a popular term "lonely rabbit" because it's believed that "a rabbit can die if it's left alone". Extending that, there's also "rabbit-type girl" which is described as "a girl who often feels lonely".