Name trends in the period
If this is right after the Vikings were a thing, they most likely just turned [name_u]Christian[/name_u]. Biblical names were on the rise.
Also, people didn’t start using middle names here until the mid 1500s, and even then it was just nobility until the 1800s. Nicknames are fair game, though.
It was rude to use god names as people names
Back when people belived in Norse mythology, it was considered rude to give people god names. People named after gods became more of a thing with national romanticism in the late 1700s/early-mid 1800s. What they would have done before that, was Norse word names unrelated to gods, and (very common) mashing up a god name with a word name. For example, they would take the god [name_m]Ing[/name_m], and then they would add another word to make [name_f]Ingrid[/name_f] (Ing+beautiful), Ingegerd (Ing+fence/protection), [name_m]Ingmar[/name_m] (Ing+excellent/famous), [name_f]Ingeborg[/name_f] (Ing+castle/defence).
Spelling nitpick
I would really recommend either avoiding names with unfamilliar characters, or spelling them with their correct characters. So you can have Synnøve (Norweigan form), or [name_f]Sunniva[/name_f] (broader Scandinavian form), or Sunngifu (Old [name_f]English[/name_f] form), but you shouldn’t use Synnove.
What I would name them:
[name_u]Pirate[/name_u] [name_f]Queen[/name_f]: [name_f]Sigrid[/name_f] “Sigge” Siøfar Asgersdatter
Reasons
[name_f]Sigrid[/name_f] is a solid name choice, so I’m keeping it and removing the middles. [name_m]Sigge[/name_m] is a very old surname for names that contain the beginning “Sig-” (meaning victory). It’s mostly used as a shortform for [name_m]Sixten[/name_m] (spelled Sigsten in ye olden times), which is a boy name, so you would get some gender bending here.
I think it’s cool that you want some sort of pirate/warrior name for her, and it does fit the setting! They just would never use a god name like that. If I were you, I would draw inspiration from historical people like [name_m]Sven[/name_m] Tveskägg (Sven Two-beards), [name_m]Erik[/name_m] Blodyx (Erik Blood-axe), [name_m]Harald[/name_m] Hårfager (Harald Hair-pretty) and make up something similar for her. I can help with translation if you want, but right now I don’t really know her character well. Possibly you could use Sjøfar (seafaring). Grammatically that would either be [name_f]Sigrid[/name_f] Siøfar or Siøfarar-Sigrid, you could pick one or switch.
Pirate’s daughter: [name_f]Thordis[/name_f] Sigridsdatter
On surnames for children of unwed mothers and also other choices.
[name_f]Sif[/name_f], [name_f]Freya[/name_f] and [name_f]Sigyn[/name_f] are all god names. I don’t know any girl names that utilise these, but [name_f]Sif[/name_f] is the wife of [name_m]Thor[/name_m], so I’m using his name instead. Some alternatives:
- [name_f]Thordis[/name_f] is a combination of [name_m]Thor[/name_m] and Dis (meaning both fog (mystical, encompasses both the seafaring thing and the poetry thing) and goddess (you mentioned that you wanted [name_f]Freya[/name_f] because she was a lady)
- Thorunn/Thorunnr is a combination of [name_m]Thor[/name_m] and Unn/Unnr (the daughter of the sea god Ægir, and also a word that means something similar to “wave”)
- Thorbjørg is a combination of [name_m]Thor[/name_m] and Bjørg (castle/defence, (same as Borg from before, but more Norweigan))
- Thorgerd is a combination of [name_m]Thor[/name_m] and [name_u]Gerd[/name_u] (meaning fence, protection) so could mean something like “please [name_m]Thor[/name_m], protect this girl”.
- Thorfhildr is a combination of [name_m]Thor[/name_m] and Hildr (meaning battle)
- Thorgun/Thorgunn is a combination of [name_m]Thor[/name_m] and Gun/Gunn (a valkyrie who rides on a wolf).
As for her surname, typically, a daugther would get the dad’s name + datter if the parents were married, even if the parents married after the birth of the child. I see that you already know this, and have come up with a solution on your own. I agree with you, but want to tell you why.
If the child was born outside wedlock, there were a few options. The most common was that the child counted as under the authority of her mother’s father, so she would get his name + datter. (This would give her the same surname as her mother). It was also fairly common for the mother to not say who the father was, and then marry someone else (in cases where the biological father was unpleasant). The child might then take her stepfather’s name + datter. Another option was calling the child by the biological father’s name + datter as a snide way to tell the community what he did.
[name_u]An[/name_u] unwed mother faced a lot of backlash from the community, and the child’s name was a good way to give him some of the blame too without revealing enough (ie outright telling everyone who he was) that anyone would force them to marry. It was also useful if the father was already married to someone else.
That being said, I don’t dislike the idea of giving Sigrid’s daughter the surname Sigridsdatter (it would need an S after [name_f]Sigrid[/name_f] for the sake of grammar), since it has become popular in feminist circles since the 1970s. Sure, it’s anachronistic, but it’s not implausible that people would have done so before. Especially since [name_f]Sigrid[/name_f] seems to be an unconventional sort of woman.
So, I think the surname should be Sigdridsdatter or Asgersdatter
Norweigian [name_f]Princess[/name_f]: [name_f]Margarethe[/name_f] Biørnsdatter, alternatively [name_f]Abigail[/name_f] Biørnsdatter
Reasons
I think a [name_f]Princess[/name_f] in this time period really needs a [name_u]Christian[/name_u] name, so that her father can show off how forward thinking and righteous he is. [name_f]Abelone[/name_f] is sadly a greek mythology name, even though it is really pretty. [name_f]Abigail[/name_f] is a close alternative. It is very uncommon here in Scandinavia, and I think it always has been. More common biblical names are [name_f]Ada[/name_f], [name_f]Hannah[/name_f], [name_f]Susanna[/name_f] [name_f]Esther[/name_f], [name_f]Rakel[/name_f], [name_f]Sarah[/name_f], [name_f]Martha[/name_f], [name_f]Eva[/name_f], and [name_f]Magdalene[/name_f]. Now that I think of it, [name_f]Margarethe[/name_f] isn’t Biblical (I think) but it was super common among richer folks around the time Christianity came.
[name_m]Bjørn[/name_m] is a modern spelling. Biørn is older, and also looks more royal to the modern eye. So Biørnsdatter it is.