Those who were born and raised in Westeros and have either First Men or Andal heritage will likely follow the most customary Westerosi naming systems for which the link can be found here.
Most of Essos and select houses in Westeros will use Valyrian naming systems that came from High Valyrian. Examples of such Westerosi houses might be Targaryen, Velaryon, and Celtigar. The link for the Valyrian name generator can be found here.
The language used in Slaver's Bay is a bastardization of Old Ghiscari. As such, they have a significantly different naming system than Valyrian. The link for the Ghiscari name generator can be found here.
Aside from the naming conventions, there is one other important factor regarding bastards. Bastards with a noble parent receive specialized last names by region. The list is as follows:
Crownlands: Waters
Dorne: Sand
Iron Islands: Pyke
North: Snow
Reach: Flowers
Riverlands: Rivers
Stormlands: Storm
Vale of Arryn: Stone
Westerlands: Hill