(Hat tip to Chuck Clark, who saw it on YouTube and asked what the odd-looking stringed instrument was.)
Here's the song, from Grieg's Peer Gynt Suite, as arranged by J.L.Lenoir of the French quartet Boann and and performed by Céline Archambeau (on vocals and harp), Lenoir (crwth), Eléonore Billy (hardingfele), and Gaëdic Chambrier (guitar):
Boann is named for a Celtic goddess associated with the River Boyne in Ireland. According to their Facebook page at https://www.facebook.com/boannquartet/, Boann is "est un quartet dont la musique est tournée vers la mer du nord, à la croisée des cultures scandinaves et celtiques" [a quartet whose music faces the North Sea and the blending of Scandinavian and Celtic cultures]. So "Solveig's Song" is right up their alley.
Solveig, who is named for a Norse goddess, is a character in Norwegian playwright Henrik Ibsen's Peer Gynt, a Romantic-era bricolage of trolls, dairy maids and other figures from Norwegian folklore. She is Peer's own true love, with whom he unites at the end of the play, and Grieg's song is especially lovely.
So is Boann's interpretation of it, and the odd stringed instruments fit right in.
The hardingfele, or Hardanger fiddle, looks like an ornate violin, but it is a Norwegian folk instrument with four or five sympathetic, or resonant, strings beneath the four strings of a standard fiddle. Played well, it has a distinctive modal sound that's perfectly adapted to Grieg's melody. The crwth (pronounced "crooth") is Welsh, not Norwegian, but it comes out of that North Sea cultural area that Boann interprets.
According to Wikipedia, which has an unusually detailed article at https://en.wikipedia.org/wiki/Crwth, the crwth was once played widely in Europe, although it was especially associated with Wales. It is similar to Scandinavian lyres of the Viking and early medieval periods. (See Gjermund Kollveit's article at http://www.musark.no/wordpress/wp-content/uploads/2013/11/The-Early-Lyre.pdf.) Whatever its provenance, it fits right in backing the other instruments in Boann's interpretation of "Solveig's Song."
There isn't much of a market today for medieval Welsh instruments, but there are people who play replicas. They're especially well suited to playing backup, but English luthier Michael J. King, who makes crwths, plays a simple melody toward the end of a video demonstrating one of his instruments.