The system's not completely unique to Iceland: after all, it's how all the common English surname
s in -son
came about. "Robert Johnson" is (in origin) Robert John's son. But you'll notice we lose one of the two S's. But in Icelandic they both stay: Jónsson. The first S is from the genitive
of the name. In English all genitives end in S. Not so in Icelandic.
Take the aforementioned Sindri. When he has children: in English we'd say Sindri's son, Sindri's daughter, with the internal S. But in Icelandic -i changes to -a, so if he has a daughter Anna she'd be Anna Sindradóttir.
The ending depends on the declension: it is not always -s. Björn becames Björns or Bjarnar. So if he had a son Sindri he'd be called Sindri Björnsson or Sindri Bjarnarson.
This patronymic system is used in various places in the world. The Bulgarian naming system is similar, and it was also used in Mongolia through most of the twentieth century.
Okay, I've been C!ed so I'd better add a word or two about Bulgarian and Mongolian. Neither uses the words son
as Icelandic does.
In Bulgarian the suffix is -ov for a son and -ovna for a daughter (and -ova for a wife). Other Slavonic languages also use similar patronymics, but in Bulgaria they are newly created surnames each generation: Todor Zhivkov's daughter would be not Marina Zhivkovna (as they do in Czech) nor Marina Todorovna Zhivkova (as they do in Russian), but Marina Todorovna.
In Mongolian the father's name with the genitive -yn was prefixed: so Tsedenbal is called in full Yumzhaagiyn Tsedenbal, but neither part is his surname. However, this system is modern, since Mongolians did have surnames but these were abolished after the Communist takeover of the 1920s in order to break the traditional clan structure. From the 2000s Mongolian were once more required to adopt surnames, preferably their former ones if these could be learned from elders or records. For more details see for example: