'Run off into the Blue'

[Mithalwen]

I don't know for certain but I guess the expression comes from the horizon appearing blue on landscapes as well as seascapes. We also have an expression "in to the wild blue yonder" which has similar meanings.

People or things who arrive unexpectedly and suddenly are also said to "come out of the blue".

Your English is fine by the way and I am sure that I speak for us all in saying you, and your questions, are very welcome and I hope you will enjoy your visits to the downs.

I don't have the Hobbit with me so I don't know whether the capital letter is original or in the translation only. If it is in the original - it may be to make clear that he is using blue as a noun not an adjective. Tolkien does sometimes capitalise things when they have special significance, it was something that they made more consistent in the revised text I think.