The verb kigkli/zw -- of which this is ostensibly the present indicative, third person singular (but see below) -- is derived from ki/gklos "dabchick" (kappa 1585), and literally means to act like the dabchick, which proverbially wagged its tail.
The wording of the entry is the same in Photius kappa695 Theodoridis, and similar in the scholia to Theocritus, Idylls 5.116 (on which see below). Other lexica add the sense "to tempt", from the same motion applied to humans (Herodian, Schematismi Homerici 114 and Hesychius give both, the Etymologicum Magnum just "tempt"). Ps.-Zonaras and the Etymologicum Magnum also give "safeguard", deriving the verb instead from kigkli/s "lattice gate" (kappa 1582).
Outside the lexica, the verb occurs in two places. In Theognis 1.303, the sense is extended metaphorically from "wag" to "change constantly": "one should not 'wag' a good life, but be unwavering". The infinitive used, kigkli/zein, which is also the form cited by Herodian and Hesychius, may have led to Photius' and the Suda's kigkli/zei.
The other attestation is in Theocritus, Idylls 5.116, a reading noted by the scholiasts as [e)]kigkli/zeu "you were wagging"; in Gow's edition it occurs as a compound verb, potekigkli/zeu from proskigkli/zw: "Why, don't you remember when I screwed you, and you were wagging your tail nicely, grinning, and held on to that oak tree?" This is clearly the sense behind Herodian's "tempt". The Photius/Suda sense is vague enough to apply to it, but "make a sound" may well be guesswork.