Divergent cognates are going to be a feature of any language family (as pointed out in this paper by Larry Trask used in the answer to this question about "mama" and "papa" words), and because of this I think this question is incredibly open ended and subjective. However, I appriciate the opportunity to share some of my favorite etymologies.
Afro-Asiatic is estimated to be a fairly old language family and therefore has a large number of divergent cognates, I personally enjoy the divergence in my own name which I found here.
- English David, from Hebrew דָּוִד (Tiberian Dāwîḏ, originally dōd), meaning "beloved", but originally a kinship term meaning "uncle"
- Mokilko ʔândé, "in-law"
- Tawllemmet idda, "Dad"
all from Proto-Afro-Asiatic *di/ad- with an unclear meaning but it was probably something like "elder".
Indo-European languages, as you and Michaelyus have pointed out, also have words with fun etymologies, and for some reason Armenian tends to show up a lot:
- English birth
- Latin fortūna, "luck, fortune"
- Armenian -աւոր (-awor), a suffix that forms adjectives, with the meaning “possessing, related to”, as in վտանգաւոր (vtangawor, "dangerous"), from վտանգ (vtang, "danger") + -աւոր
all from PIE *bʰer-, meaning "to bear, carry" (English bear is also a decendant).
This one is kind of cheating, because it's a borrowing, but Japanese 蜜 (mitsu, "honey") and English mead are genetically related through Tocharian B mit from PIE *médʰu. They're all kind of similar looking, I know, I just think it's cool.
Finally, I don't think anyone can discuss divergent cognates without mentioning John McWhorter's famous comparison of Algonquian words for winter:
- Cheyenne aa', "winter, to be winter"
- Ojibwe biboon, "winter, it is winter"
From the Proto-Algonquian *pepo·nwi, also meaning "it is winter".
Obviously, this is not an exhaustive list, there are probably thousands more cognates with this level of divergence because languages do weird stuff all the time.