The Sälçüks (Kharakhanid: سَلجُك; Persian: سلجوق) were a dynasty of Oğuz Turkmän origin, founded by members of the Qınıq tribe. They are one of the ancestors of the Türks of Türkiye.
Sälçük is the form given by Mahmud al-Kashgari in his Dīwān Lughāt al-Turk or "Compendium of the Turkic Languages." There, it is transcribed as Sälçük (سَلجُك) Kharakhanid *sält͡ʃyk. There is also the variant, Salçuq, which is featured in Persian transcriptions.
Given Mahmud al-Kashgari's expertise, it is safe to say Sälçük is the correct form. Unfortunately, it is of uncertain etymology. Though, it has been claimed that it is derived from Arabic sayl (سَيْل) /sajl/ or "flood" and the Turkic diminutive suffix *-çük. Yet, an Arabic etymology seems unlikely given Sälçük Bäg's personal history. As warriors under the Qazar Qağanate, both Sälçük and his father, Duqaq, were heavily influenced by their Judeo-Oğuric overlords. So much so that Sälçük would give his children Judaic names. Moreover, their association with the Qazars would have made them hostile to the Arabs, providing little incentive for Duqaq to name his son from their language. Given that Sälçük's name is not Judaic, one would then assume it must be Turkic.
If it is derived from an Oğuz language, then there are no obvious etymologies. However, if one considers an Oğuric etymology, then there is Oğuric *s(ʲ)äl or "wind." It is comparable to Mongolic *salkïn s.m. (Nugteren ,2011) and survives in Çŏvaş as ҫил /sʲäl/ s.m. It is also a loan in Hungarian as szél /seːl/ s.m. Thus Sälçük could mean "little wind."