Names are complex, and quite culturally variable in the real world. The notion that your name needs to be "organized" is comparatively recent, and the idea that it has a fixed number of parts is more recent than that, and not followed in all cultures. (I'm not even sure it's followed in "most" cultures today.)
Anyone who is dealing with name systems should read
Falsehoods Programmers Believe About Names. Read, or at least skim, the comments as well; some useful ideas and examples have accumulated there.
Even if you're limiting your search space to historical European-style names, you find out that it's weirder than you realize... "Nicholas If-Jesus-Christ-Had-Not-Died-For-Thee-Thou-Hadst-Been-Damned Barebon" was a
real person (basically the inventor of fire insurance).
Maria II of Portugal was born as "Maria da Glória Joana Carlota Leopoldina da Cruz Francisca Xavier de Paula Isidora Micaela Gabriela Rafaela Gonzaga", then ended up queen, betrothed to someone who used it as a ruse to start a coup, married after the restoration to someone else who died quickly, and then married again to a third person. One of her kids was named "Maria Ana Fernanda Leopoldina Micaela Rafaela Gabriela Carlota Antónia Júlia Vitória Praxedes Francisca de Assis Gonzaga"... but after *her* marriage was generally known as "Princess George" as I understand it.
And there are even places where "permanent" family surnames are not only not the norm, but nearly forbidden; look at modern
Icelandic names. This is particularly relevant for DF, as many people tend to think of dwarves as generically Nordic, and Iceland has preserved something much closer to the historical Nordic system than most modern countries.
The Icelandic system is by *default* patronymic; Jon Einarsson has a son Olafur Jonsson and a daughter Sigridhur Jonsdottir. (I'm heavily simplifying the accent marks and non-standard characters; and handling some things such as professions as their English equivalents.) However, it's legal to go matronymic, use as the base the father's middle name, have a dual patronymic-matronymic name, and a few other variations.
Of particular interest is that in cases where there is ambiguity, you can either go up another ancestry level to add a grandparent, or add a profession. Phone books are alphabetized by given name, surname (patronym, etc.), profession. So, if Jon above was a smith, he might be Jon Einarsson Smith; if his son also took up the profession, he might be Olafur Jonsson Smith; but if he apprenticed somewhere else he might be Olafur Jonsson Baker. *His* son could be Hallur Olafurson Smith only if he took up his grandfather's profession; he might be Hallur Olafurson Fisher, or decide to go by Hallur Olafurson Jonssonar (son of Olafur, grandson of Jon).
Also of interest to DF is that the list of names is mostly fixed; you can't just make up a first name without going through approval of a committee, and surnames have to be based on one of the existing formula. Ideal for computer generation; a fairly simple set of names and rules could generate the vast majority of legal Icelandic names.