Agusta

Birna

A top name in Iceland, while its parent name Alvar, ranks highly in Sweden and Finland.

Elvar

Emilia is the feminine form of the Roman clan name Aemilius, which derived from the Latin aemulus, meaning "rival." In Shakespeare’s Othello, Emilia is the wife of Iago and confidante of Desdemona. Amelia, although homonymous, has a different root and meaning. 
 
Emilia is rising as an Emily/Amelia alternative. In modern pop culture, Emilia is the name of the main character in The Princess Diaries and the first name of Game of Thrones and Solo star Emilia Clarke.
 Emilia is currently enjoying high popularity across some diverse cultures—it's near the top of the charts in Germany, Finland, Austria, Poland and Chile, and is favored in both English- and Spanish-speaking countries.

Emilia

Erikur

The dynamic personality and paintings of Mexican artist Frida Kahlo have inspired growing numbers of parents to resurrect this form of the name. It does much better in certain European countries, especially in Denmark and Norway. The Frieda and Freida spellings were more popular in the US until the middle of the 20th century.

Frida

Freydís

Guðmundur

The Old Norse word for "harp", also designating the constellation Lyra (which means "lyre" in Greek). It's popular in Iceland, where it is also the name of the first month of summer in the old Icelandic calendar.

Harpa

Halldór

Haukr

Johanna is the version of this name used in Holland, Germany, and Scandinavia. The extra h makes Johanna a slightly more dignified version of Joanna. 
 A notable namesake is the author of Heidi, Johanna Spyri. There are Johannas in Sweeney Todd, The Hunger Games, and the Bob Dylan song, "Visions of Johanna."

Johanna

Jonas has a slightly more grandfatherly image than the English version of his name, but that only adds to its retro appeal. And though it may lag behind Jonah in this country, Jonas is riding a huge wave of popularity in Europe, where it ranks highly in Germany, Switzerland, Austria and Norway.
 Over time, it has been associated with Dr. Jonas Salk, developer of the anti-polio vaccine, and tween fave group the Jonas Brothers. Charles Dickens used the name for a character in Martin Chuzzlewit.

Jonas

Katrín

This is an alternative to Laura or Lauren made romantic by Dr Zhivago, and badass by video-game heroine Lara Croft.

 In mythology, Lara was a nymph in Ovid's Fasti known for her beauty and inability to keep secrets. As a place name, Lara is a state in Venezuela.

Lara

Lilija

Margrjet

Rakel

Reynir

Sóley is the Icelandic word for the buttercup flower, derived from the words meaning "sun" and "island." It is also used on the Faroe Islands as a short form of Sólja.

Soley

Sigríður

Svartur

The most common form of Thomas in various European languages, although some Anglophone parents also use this as a phonetic spelling of Thomas.

Tomas

Vestmaðr

Veturliði

Viktoría

Ágúst

Álfsól

babettee's list "Cool Icelandic Names" of 28 great name ideas: Agusta - Álfsól!

Cool Icelandic Names

Cool Icelandic baby names and their meanings. Iceland has a rich culture, which comes through in the naming style of Icelandic parents!

So, take a look at my list and choose your favourite!

Note:
Iceland is very strict about baby naming laws.
There are 1853 approved names for girls and 1712 approved names for boys. If you reside in Iceland, to choose a name outside of this list, you must seek special permission. The authorities will see if the name is compatible with the Icelandic tradition or not.

NOT ALL OF THESE NAMES ARE ONLY SEEN IN ICELANDIC CULTURE! SOME HAVE BEEN ADOPTED FROM OTHER CULTURES.