Considering this questions is quite outdated, as well as its answers, I decided to open a new, more targeted one.
As some of you may know in League of Legends has been released a set of Christmas-themed summoner icons that can be unlocked by players who fulfil certain criteria, such as never having been banned or having entered a queue with a 5-man-team during the Christmas period.
So, what are the exact conditions required for every one of the available icons?
Unfortunately Riot never officially published the details on how to unlock them. However if we are to believe the various rumours around related forums and the internet, the image below is a very nice compiled list of hints. I myself can confirm the accuracy of at least the first hint as I got it only days ago.
Doran’s Wreath Icon: Enter a queue as part of a 5-member premade team.
Stocking Blades Icon: Buy RP as a gift for a friend.
Trimmed Turret Icon: Buy a Champion as a gift for a friend.
Reindeer Urf Icon: Buy a skin as a gift for a friend.
Holiday Nashor Icon: You will have this earned at the beginning of the Snowdown period if your account has never been banned (or received a warning?)
Shopkeeper Icon: This icon will be awarded at the end of the Snowdown so no-one is really sure about the conditions. Most seem to agree though that based on the hint one must have unlocked a certain amount (perhaps even all) of the other 5 Snowdown summoner icons in order to earn this one.