The headword is masculine nominative singular of the adjective klu/menos, cognate with the verb klu/w (cf. kappa 1841, kappa 1842, kappa 1844 and LSJ at web address 1). The first part of the entry seeks to unpack this etymology: see next note.
[1] From ancient commentary on Callimachus, who uses the headword in the genitive case, and in the upper-case way noted by LSJ s.v.: see Hecale fr. 285; also fr. 668 in the fragmenta incertae sedis. Note that LSJ also characterises Klymenos as an underworld deity in his own right; cf. Pausanias 2.35.4.
[2] Greek Anthology 7.9.7-8 (ascribed to a Damagetus), web address 2; 1071T in Bernabé et al.; cf. Gow and Page (vol. I, 77) and (vol. II, 224-225). See already at alpha 864 and further extracts from this epigram at alpha 3117, alpha 3204, and pi 2515.
[3] Greek Anthology 7.189.3-4 (ascribed to an Aristodicus Rhodius), web address 3. On this epigram, an epitaph for an a)kri/s (locust, cricket, or grasshopper; cf. alpha 985), see Gow and Page (vol. I, 42) and (vol. II, 108-109). Gow and Page note (vol. II, 107) that nothing is known of this epigrammatist. However, they observe that he would appear to be an imitator of Anyte (ibid.) and indeed that the Anthologia Planudea (Codex B) ascribed the epigram to the poetess (vol. I, 108-109).