I respectfully question the etymology given for this name. Tonight, I spent quite some time trying to search the Internet – especially Turkmen dictionaries – for any words akin to nyazik or else meaning “graceful”; but I failed to find any viable sources. The Turkmen one word I did, in fact, was näzik meaning "delicate, gentle; tender, soft, sensitive”. This word is a descendant, via Ottoman Turkish نازك (nâzik) “delicate, easily damaged”, of Persian نازک (nâzok) “delicate, tender” ). [noted -ed](Like with many Semitic and Semitic-influenced languages, these words all have many more meanings and senses that I intentionally didn't mention here, for brevity's sake; however, links to Wiktionary's entries for these words are: Nâzok, Persian: https://tinyurl.com/PerNazok (scroll to the bottom to see a "family tree" of the word, including Turkmen näzik - an individual Wiktionary entry for näzik hasn't been created yet). Nâzik, Ottoman Turkish: https://tinyurl.com/OTurkNazik. Nazik, Azerbaijani/modern Turkish, https://tinyurl.com/AzTurkNazik
(Like with many Semitic and Semitic-influenced languages, these words all have many more meanings and senses that I intentionally didn't mention here, for brevity's sake; however, links to Wiktionary's entries for these words are:
Nâzok, Persian: https://tinyurl.com/PerNazok (scroll to the bottom to see a "family tree" of the word, including Turkmen näzik - an individual Wiktionary entry for näzik hasn't been created yet).
Nâzik, Ottoman Turkish: https://tinyurl.com/OTurkNazik.
Nazik, Azerbaijani/modern Turkish, https://tinyurl.com/AzTurkNazik