Apparently, the gift is really a promotional stunt for their upcoming Big Family Games.
"BIG Family Games is the ultimate Wii game to get all family members, from grandparents to young children, playing together. The Royal Family is arguably the most important family in the country so we felt that they had to have a copy of the new game. But we thought that Her Majesty the Queen wouldn't want to play on any old console, so an extra-special gold one was commissioned. We hope that she and the rest of the Royal Family enjoy the game!"
Yes, I'm sure the Queen will be all over this after a few lessons on these new things called "video games" from her grandchildren. [Kotaku]