Is that how it works? I thought it was based on parentage not geography.
Here's what the GoT Wiki has to say on the subject under the Terminology section:
The stigma of illegitimacy is so great that all acknowledged bastards born to a noble in Westeros have to identify themselves through a specific surname marking them as a bastard, which varies by region:
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.
- Flowers: The Reach
- Hill: The Westerlands
- Pyke: Iron Islands
- Rivers: The Riverlands
- Sand: Dorne
- Snow: The North
- Stone: The Vale of Arryn
- Storm: The Stormlands
- Waters: The Crownlands
Which possibly invalidates my statement that Jon is technically a Sand; according to that information, he may have been called either Jon Snow OR Jon Sand depending on whether or not it was decided that his mother's homeland (The North) trumped where he was born (Dorne).