According to some fans, the 'Game of Thrones' title sounds better than the other title 'A song of Ice and Fire', for it's shorter and easier to recall.
Some people believe that the shorter and the easier name was chosen because it's better for the branding of the show.
The title 'Game of Thrones' provides a much better explanation to the show than the other title 'A Song of Ice and Fire' which can be very vague. Since the show is about politics and fights over thrones, the first title is much more accurate.
Game of Thrones was the title of the first novel written by George RR Martin in the series. It made sense to HBO to use the title of the original book for the series.
The title 'A Song of Ice and Fire' won't be understood by people who haven't read the novels. The title will be understood later on as the show proceeds. This is why it made sense to use the title 'Game of Thrones' instead. See why some people dislike Game of Thrones.
The title 'Game of Thrones' is a better match to the medieval environment the show takes place in. The title 'A Song of Ice and Fire' would have made the show sound more like a fantasy show than a deep political one.
According to some fans, the selection of the title could have been made to avoid spoiling the future events that will happen in show. See why is Game of Thrones popular.