Elvar started his playing career with Njarðvík in the Icelandic top-tier Úrvalsdeild karla in 2011, averaging 11.5 points and 4.6 assists during the regular season.

[3] Following his strong performance, Elvar agreed to join Long Island University where he played alongside Martin Hermannsson[4] with the LIU Sharks basketball team.

[12] On 5 November 2018, it was reported that Denain Voltaire planned to release him to make room on the roster for another signing.

[15][16] On 13 December 2018, Elvar posted a triple-double in a victory against Breiðablik with 40 points, 11 rebounds and 12 assists.

[19] In March 2020, he won the Swedish championship after the Swedish Basketball Federation decided to cancel the rest of the 2019–20 season due to the coronavirus outbreak in the country and award Borås the title as the top team at the time of the cancelation.

[23] On 21 April 2021, he had 33 points and 11 assists, while shooting a perfect 9-of-9 from the field and 10-of-10 from the free throw line, in a 103–90 victory against BC Pieno žvaigždės.

[24] On 7 May, he had 15 points and a season high 17 assists in a double overtime loss against BC Neptūnas.

[45] On 13 August 2021, Elvar scored a game high 30 points in a 91–70 victory against Denmark in the 2023 FIBA Basketball World Cup Pre-Qualifiers.

[47] Elvar is the son of former Icelandic national team player Friðrik Ragnarsson and brother of former Úrvalsdeild point guard Ragnar Helgi Friðriksson.