Updated June 2011: Kit Harington is doing a great job as Jon Snow - very happy with him. At first I thought it didn't quite match as Jon wasn't supposed to be this strong looking guy, but Kit is doing such a fantastic job that he's grown on me as Jon Snow.
When I was reading the books, I imagined Jon as Zac Efron, who is known for being in those corny High School Musical movies.
Jon Snow is a the claimed bastard son of Lord Eddard Stark, by a mother whose identity is not widely known. His direwolf companion is the mute, albino Ghost (due to his albinism and eerie silence, cementing the 'outsider' status caused by Jon's bastardy).
Jon, who received the surname customarily assigned to illegitimate northerners, is of an age with Robb. He was sired during the War of the Usurper, and Ned took possession of him some time after the battle at the tower of joy, which only Ned and Howland Reed survived. Despite his wife Catelyn's objections, Ned insisted on taking responsibility for the boy, and he grew up side-by-side with the trueborn Starks. He has a long face, grey eyes and lean build, with strong Stark coloring—in fact, he looked more northern than many of Lady Stark's trueborn children, which was part of why Catelyn could never accept his presence in her household. Unlike the more usual experiences of illegitimate children, Jon knows the identity of his father but has no idea who his mother might be.
As Jon was growing up, his position became increasingly awkward. During King Robert Baratheon's visit to Winterfell, Jon asked his uncle Benjen Stark to join the Night's Watch and was refused at first but was eventually allowed to go. Jon was initially resented by other recruits for his noble background, but he learned to fit into the crowd by assuming leadership and uniting the recruits first against their caustic master-at-arms, Alliser Thorne, and then in protection of the cowardly but good-natured Samwell Tarly. In the beginning he had trouble adjusting to the life of the Night's Watch, as it was very different from what he had imagined. At the outbreak of the War of the Five Kings, he tried to desert to join Robb's army, but his new friends brought him back before he could be punished for desertion. By finally deciding to honor his bonds and abandon his past, Jon came to accept the Night's Watch as his new brothers.
Jon Snow is my favourite character in the Song of Ice and Fire series and the guy I picture as Jon is Zac Efron.