We've identified the problem!
For the record...@ladyscorpio is correct
"
Bastard surnames are dependent on the region a child was born in, i.e. where the mother is from, not where the father is from. For example, a noble lord from the Stormlands could father one bastard child in the Vale, and another in the Riverlands, but neither would use the surname "Storm": the first bastard would use the surname "Stone", and the second would use the surname "Rivers." It is extremely unusual for a bastard to know who his nobleman father is, but not his mother. Therefore
Jon Snow's situation is additionally unusual, not just because he actually lives with his nobleman father, but because he wasn't even born in the North.
Eddard Stark brought him back to
Winterfell as an infant after fighting in the south during
Robert's Rebellion, but refused to say who his mother was or where she came from. As a result of the mystery surrounding his mother's identity, Jon ended up using the surname "Snow" by default."
http://gameofthrones.wikia.com/wiki/Bastardy