And How Shall I Call You m'Lady?
You'll probably pick a name fitting to your culture, although there are exceptions. Crusaders, Vikings and Mongols brought home names from a variey of places. Your first name is most personal, and is largely up to you and your imagination. People often choose first names that describe their personalities, that have symbolic meaning, or simply sound nice. Some people join knowing just what they want to be called, perhaps an ancestral name, or one they always wish they had been given. Others try out several names before taking a permanent name. If you need help, library reference departments have books that can tell you what names mean and foreign or ancient forms of them. The web is also a good resource for this information.
There are a number of traditions to follow for surnames, and you can use more than one, either together, or as alternatives. Hereditary family names were passed on by some noble families as early was 1000 AD, but most of our ancestors didn't receive family names until several hundred years later. Provincial Danes resisted the idea until World War I and Icelanders still don't have them. Your surname, or names, are more likely to describe you as a individual, than your entire family.
While trying to decide on a permanent Society name, you might consider the names that describe you:
1) geographically. Historically, this would be the name of the farm, town or province you, or your ancestors, came from. Examples are: av Gotland; av Dverell; Lindholdt. These days it can also be the pet name you've given your house or apartment: of Woodsholme; of Wizardskeep.
2) by a characteristic or achievement. You have to be careful with these, or you may emphasize something that would be best forgotten. Examples, good and bad are: the Quiet; the Far-Travelled; Silvertongue; Goatsbeard; the Gross; Silverhair: the Improbable.
3) by your profession, either modern or in the Society. This isn't particularly common, as many of us have professions which aren't mediaeval, or which change frequently. Two examples are: "the Printer" and "the Sarabite"
4) by social group. The leaders of many households use their household name as their surname, and sometimes all so as a show of unity. Examples are: of the Blackwater; Van Dag; of Domensque; of Abstranon; of Drachenhalle.
5) symbolically, or from fantasy. This offers the most freedom of expression, as you can choose names with obscure meanings or literary referenes, or which simply sound nice. Examples are; Steorra of AEfen (evening star); Ankestjerne (anchored to a star); Hlio Songrene (evergreen hillside).
6) by parents or grandparents. This was common in all cultures, and each language has its own forms. Some names referring to parents are: Jenssen or Jensdottir (Norse); AElfreding (Saxon); Dominquez (Spanish); Rodriques (Portuguese); MacGregor - for a son, NickGregor - for a daughter (Scots).
Now you've got a half-dozen names to choose from, and you and always make up more, or translate them into other languages. Decide which ones fit the kind of personality you want to portray, which ones sound the best together, then try them out on your friends and see if they can pronounce them. Later you'll see how many new ways the scribes can find to spell it.