What was the reason behind the song?

No explanation is ever given for the importance of the song, or its significance (if it has any). It is meant to represent the secrets that Angela Lansbury's character stole.
This film is a remake of an old Hitchcock film, and using an element like that is a Hitchcock trademark. He referred to it as a "McGuffin" (taken from an old joke). It is something very minor that is often never fully seen or explained, but it is central to the plot.

SPOILERS:

I only caught the end of this remake on TV, but the original made quite clear that the tune (which was entirely different in the original) contained a coded message (how the music translated back into words, or what was in the message was never explainted, but the point was that Miss Froy was a spy and was carrying secrets with her that were encoded in a tune. It was passed to her at her hotel at the very beginning of the journey, but someone playing the tune beneath her balcony (and who was promptly thereafter assassinated). The original also took place in an imaginary easternish or balkanish European and the political intrigue in the plot was not specified - but in the atmosphere of the late 1930's, I am sure that everyone in the British audience knew very well whom Hitchcock was referring to - so that it was an obvious revision in the remake to just openly make the bad guys into Nazis.

[deleted]