Anna's beautiful grey-white Fjord horse first appeared unamed in F1:
We learned in the book A Frozen Heart that he was a male named Kjekk, which means "handsome" in Norwegian. And handsome he is!
According to Dangerous Secrets, Iduna bought him from a farmer. Here's how the scene goes:
And here's where the line is starting to blurr. The farmer says the horse name is Havski, which also means "handsome", but in Northern Sàmi this time! And then Iduna switches to Norwegian and calls him Kjekk. We could assume she preferred to go with a shorter name so she renamed him in the scene. Except that the name Havski reappears in later books: Forest of Shadows, Polar Nights and All is Found, in which Kjekk/Havski is described as being Anna’s childhood horse.
We know the Northuldra tribe, from which Iduna comes from, is based on the Sàmi people. Arendelle on the other hand is either Norway or a part of it.
What if the farmer had Northuldran connections and subtly wanted to convey it to Iduna, to show her she's not alone, like he suspected she wasn't Arendellian? After all he knows the meaning of "Havski". But Iduna gives the horse a Norwegian name instead, maybe because she doesn't want to raise suspicions about her origins? It can't be a coincidence for the 2 names of the horse to have the same meaning.
So here's my guess: Iduna must have shared the two names with her daughter and Anna either called him Kjekk or Havski ever since. The fact that the 2 names appear in Dangerous Secrets make them both canon for me. It's not such a stretch for a horse to have 2 different names - happens also in real life depending on the owners.
So, are you team Kjekk or team Havski?