Have the directors ever explained why the TV show Game of Thrones is named after the first book, instead of the book series "A Song of Ice and Fire"? However, it is based on the whole book series.
A Song of Ice and Fire really sounds quite fantasy-ish, whereas Game of Thrones sounds medieval and more suitable for a show. Also, the name 'Game of Thrones' gives more information about what will happen in the TV series (politics and fight for the throne) than the name A song of ice and fire does.
Game of Thrones is still a very suitable title, yes it is the title of the first book but the entire span of the series covers the game being played for the Iron Throne.
A Song of Ice and Fire is sophisticated and poetic, but Game of Thrones is direct and exciting and very easily remembered. And that's what counts for the average person. Note also that the phrase is used in the other books and is perhaps even better than the series title at describing in one phrase the whole point of the books.
The word "game" is active where the word "song" is passive. "Game" implies conflict, struggling for position, winners and losers. "Ice and fire" are very evocative images, but they are also indistinct in that they don't really suggest what the song is actually about, except that fire opposes ice. "Thrones" evokes a world of kings and courts. Taken as a whole the phrase "A game of thrones" vividly calls to mind intrigue and wars of succession.
TV audiences have a short attention span. They don't have enough time to read a 6 word titles, 4 is enough.
More importantly, A song of ice and fire refers to:
The eventual fight between the White Walkers and Mellisandre, Stannis et al, which I very much suspect will receive a lot more attention in later books/series. The game of thrones that is taking place is unlikely to be all that important directly to the final plot (apart from it's impact on Stannis and the Night's Watch), but it appears important immediately.
So we have a title that takes a long time to say and become relevant, vs a title that is relevant immediately.
For a book series, you choose the title that is more relevant, but without the immediate gratification and with the potential to cause confusion/mystery for a while as people don't know why it's relevant.
For a tv series, you choose the title that is snappy, relevant from millisecond 1 and avoids any confusion/mystery at all costs.