On this website, it is mentioned that Kai might be considered as a Frisian diminutive of Gerhard, Nicolaas (Nicholas), Cornelius, or Gaius.
I can see the relationship between Kai and Gaius (Caius, hence Kaius, Kai), but I wonder if there is any link between
The etymological dictionary of Dutch and Frisian names (dr. J. van der Schaar, Woordenboek van voornamen) mentions "Kei" (pronounced "Kai" in some dialects) and its variants "Kaei, Kaey, Kay, Key. Keije, Keijen" and says they are Frisian and are diminuatives (he calls them "vleivormen", so affective (positive) names, used in the speech of siblings, e.g.) and refers to names "Gerrit/Gerhard", "Nicolaas" and "Cornelis". So the author considers that these names could have derived from any of those traditional names. He offers no definite etymology beyond this. As Yellow Sky said in his answer, such forms are often quite far removed from the their original forms. The Danish name "Kaj" is mentioned under "Gaius". "Gaius" is not mentioned as the origin of "Kei", though. He does mention that maybe the Danish Kaj could have the same origin as the Frisian "Kei", but the Danish can certainly be derived from "Gaius" as well.
I think that if the name Kai is given nowadays, it's probably because people know the Danish name, which I think is more well-known than the traditional, but rare, "Kei"-forms at the beginning. This is just my impression, though.
Very often diminutives of personal names bear no resemblance to the original, consider the Russian Шура [ˈʃurə] which is a diminutive of Александр (Alexander), or the English Dick for Richard, or the Spanish Chucho for Jesús. Such diminutives not readily explainable, they are just a convention.
rcould have been dropped. Also, I doubt the name is Frisian as it doesn't occur in the Netherlands. My guess would be that it's Danish in origin and derived fromKallebut a better judge of Danish phonology would have to say more about that.