At some occasions in Game of Thrones now and then Tyrion Lannister uses to whistle some melody (at least during the first two seasons, where he still had a bit more reasons to be elated). Is this just some varying random tune he whistles without any further significance or is this a particular melody known to him and others (and to the book readers, if even true to the source material in this regard)?
Do you mean The Rains of Castamere?
Here's an instrumental version. It's also sung in Season 3 and 4, there's an episode named after it, and it plays during the Purple Wedding. One instance of the whistle can be found here (albeit a bit out of tune):