I have read up to the third book, so the answer may be far ahead in the series, but I have always wondered if there is a reason or a story on the vast lore that explains why giving the last name Snow to illegitimate sons, like Jon Snow or Ramsay Snow...
Bastards from the North are named "Snow", but those from other areas have different surnames, summarized here: http://awoiaf.westeros.org/index.php/Bastardy#Surnames
Region Surname Example Character Reach Flowers Ser Robert Flowers Westerlands Hill Joy Hill Iron Islands Pyke Wex Pyke Riverlands Rivers Walder Rivers Dorne Sand Nymeria Sand North Snow Jon Snow Vale Stone Mya Stone Stormlands Storm Ser Rolland Storm Crownlands Waters Aurane Waters
In legal terms, a "base-born" child, or bastard, is someone whose parents were not legally married at the time of their birth. A person whose parents were married when they were born were considered legal childen, or "true-born." Since the inheritence laws of Westeros are built around inheritence by birthright, it became neccesary to distinguish a nobleman's legal childen from his illegal children.
Therefore, all bastards of noblemen in Westeros are given a special surname to reflect that they are baseborn, and to distinguish them from someone's legally-born children. Bastards with no known relation to a noble house have no surname, like other smallfolk.
As far as why it's specifically "Snow," custom decrees that bastards are given a surname based on the region in which they are born. The names generally reflect the geography of the region. For the North, it's "Snow." Parents may give a bastard a different surname if they choose, however.
It's interesting to note that "natural childen" (a polite way of saying "baseborn") are given "natural names."