Kevlar sounds like a chemical, a Greek word or a Slavic surname. As far as the sources show, it is none of those.
The fiber came first. In 1964 a DuPont research group led by chemist Stephanie Kwolek began looking for a lightweight, strong fiber for tires, partly because of fears of a gas shortage. In 1965 Kwolek made a cloudy, thin solution of a polymer that she had trouble dissolving. At her urging, a technician, Charles Smullen, tested it on the spinneret anyway. The fiber did not break the way nylon did. After about six years of development it reached the market in 1971, according to the University of Bristol's chemistry pages.
The name came later. Wikipedia says the material was first called Fiber B and was named Kevlar in 1973. Etymonline instead dates the trademark to the fiber's development around 1965, so the exact year of the name is not settled. Its chemical name, poly-paraphenylene terephthalamide, was never going to fit on a label.
Where "Kevlar" itself comes from is the part nobody has documented. Wikipedia calls the term "of obscure origin and likely created arbitrarily." Etymonline describes it as "probably an invented word of no etymology." The Bristol page confirms only that it is a DuPont trademark. None of the sources read name the person who coined it or explain the choice.
Other DuPont names do have recorded coinages. Etymonline reports that nylon came in 1938, according to DuPont, from a "random generic syllable" plus the fiber ending -on. Teflon, registered in 1945, joined syllables of its chemical name to an arbitrary -on. No comparable explanation exists for Kevlar, and this page does not infer one from the pattern.
For now the safest summary is the one Etymonline gives: probably an invented word with no etymology.