From a comparative of cheir and teino (to stretch); to be a hand-reacher or voter (by raising the hand), i.e. (generally) to select or appoint -- choose, ordain.
see GREEK cheir
a. properly, to vote by stretching out the hand (cf. Xenophon, an. 3, 2, 33 ὅτῳ δοκεῖ ταῦτα, ἀνατεινάτω τήν χεῖρα. ἀνετειναν ἅπαντες).
b. to create or appoint by vote: τινα, one to have charge of some office or duty, passive, 2 Corinthians 8:19, and in the spurious subscriptions in c. with the loss of the notion of extending the hand, to elect, appoint, create: τινα, Acts 14:23 (see examples from the Greek writings in Passow, under the word, p. 2440{a}; χειροτονεῖσθαι ὑπό Θεοῦ βασιλέα, Philo de praem. et poen. § 9; (βασιλέως ὕπαρχος ἐχειροτονειτο, de Josephus, § 41); Josephus, Antiquities 6, 4, 2; (7, 11, 1; of the choice of Jonah as high priest, 13, 2, 2; cf. Hatch in Dict. of Chris. Antiq., under the word, Ordination, p. 1501{b}; Harnack on 'Teaching' etc. 15, 1 [ET]).). (Compare: προχειροτονέω.)