I recently encountered the name
which has been circulating for about 10 years, but
In terms of the A Song of Ice and Fire novels and the Game of Thrones tv series, the only correct spelling is khaleesi. I suspect that the name has now been adopted by people who have only ever heard it in passing. Not everyone has read all the novels so far (twice!) and/or watched the entire series. There’s nothing unusual about different spellings of a name existing side by side. What’s more unusual is that a name has a definite origin and an official spelling.
I also suspect that the drift is/will be from kh to k and maybe we’ll even see Caleesi in the near future. Using k instead of c and kh instead of k is a deliberate choice, which I am middle-aged-ish and grumpy-ish enough to thoroughly dislike.
After season 8 of Game of Thrones, the parents of all the Khaleesis and Daeneryses might be regretting their choice, or maybe they’ll pretend that season 8 just didn’t happen.
I also suspect that Daenerys is going to get alternative spellings. I spelled it wrongly the first time. Most people on discussion forums refer to her as Dany, to avoid the problem. Arya might end up as Aria, which existed already anyway.
For what it’s worth, there are about 1.2 million search results for “kaleesi” (in quotation marks), with the question ‘Did you mean “khaleesi”’ and about 8.4 million for “khaleesi”. But then again, khaleesi virus comes up as one of the suggestions.
Note that female names tend to be adopted more than male names: there are probably no Khals or Drogos out there.