Running into this Moor Next Door post on the spelling of Qadhafi's name, and this Atlantic report that his passport spells it "Kathafi", reminded me of a meeting I a few months ago. I was meeting with a bunch of business people who know no Arabic and little about the Middle East. The conversation turned to Libya and one of them turned to me and asked why there were so many spellings of Qadhafi's name. What follows is what I said, which is very much what Kal of TMND argues, except I put it in laymen's terms, without the phonetics.
In Arabic, Qadhafi's name is spelled القذافي which if you drop the article, means
ق - ذ - ا - ف - ي or q - dh - a - f - i. The "q" letter is almost unique to Arabic (sometimes called "the language of the qaf" — sorry, it's the language of the dhad, not qaf!) and often transliterated as a "k", since its pronounciation can be difficult for non-Arabic speakers. It is standard in classical Arabic and places like Fes in northern Morocco, but northern Egyptians, urban Syrians and others often pronounce this letter as a glottal stop, while southern Egyptians and Bedouins most often pronounce as a "g", as in "go". (This is why in Syria upscale Damascenes call the regime "the government of the Qaf", because pronouncing the letter is a country bumpkin thing to do, and Eastern Sunnis and Alawites — long dominant in the regime — often do it). Hence you see Qadhafi, Kadhafi or Gadhafi. The "dh" sound also has no equivalent in many languages as a standalone letter, and to top it off is made emphatic by a shedda — a kind of accent that indicates the letter should be doubled, which is why academics use the unwieldy "Qadhdhafi." And the "dh" is often not pronounced as such — in most colloquial Arabics, it is pronounced "d". I'm not sure why it might be pronounced "th", but perhaps this was used in Qadhafi's passport because it is close to the English sound in "the", which sounds very much like "dh".
I always write Qadhafi because it's simple and faithful enough without being completely anal, like Qadhdhafi.