In my fiction writing, I use the following naming convention:
First name,
Any possible nickname,
Surname (which only immediate family, including grandparents, will share--if they don't have any living immediate family, the surname is omitted. Aunts, uncles, brothers, and sisters can have different surnames, if they take their spouse's name. The spouse will then change *their* surname to their occupation. All children will inherit the surname of the parent with the occupation. So if Sally Jones marries Tom Berry, and Sally's a baker while Tom is a peasant, then Tom's last name will become Jones, and Sally's last name will become Baker. Their kids will have the last name Jones. If both parents have an occupation, they will both change their surnames to their respective occupations, and it is then permissible for any children to choose a surname of their liking, from the two (typically, the name that's been around the longest is chosen), or to create a new surname. Apprentices with no family name of their own may also adopt the surname of their masters, if their master doesn't have any children of their own, and gives permission.),
Family name (which will be-or obviously reflect- one of their parents' occupation. Sally and Tom's kid will have the surname Jones, and the family name Baker. The child of a peasant, an orphan, or a slave, will have no family name.),
Clan name (made up of two nouns--when a new King/Queen comes into power, the King/Queen takes one of the nouns from their name, and one of the nouns from their spouse's name, and forms a new Clan. Their children will become part of that new Clan. This ensures that all Clans are technically of equal political standing, since they all are of the blood of Kings/Queens. This also helps check the power of older, more established Clans, as well as the power of the King/Queen, since they'll now be the head of, and genesis of, the newest-and politically weakest-new Clan, with no legal ties to their original Clans, and nothing binding them to any other Clan, or any other Clan to them.). The Clan name is considered the most important name, but can politely be omitted in introductions. Clans never intermarry, though, and never marry anyone who shares one of the nouns in their name. Only dwarfs who don't actually know their Clan name, don't belong to a Clan, and if such a dwarf marries another dwarf with a Clan name, the Clan typically adopts them--although they'll investigate to find the dwarf's true Clan, if it seems possible to do so. No married dwarf ever lacks a Clan name, either their original one, a new one (in the case of royalty), or their adopted one. Marriage can't legally occur between dwarfs without both possessing a Clan name.
and then Fortress name (which can be either the Fortress the dwarf is living in now, or the Fortress they were born in, the choice being up to the dwarf in question.), which is never omitted, unless the dwarf has never actually lived in a Fortress.