μεσιτεύω: 1 aorist ἐμεσίτευσα; (μεσίτης (cf. Winer’s Grammar, p. 25 e.));

1. to act as mediator, between litigating or covenanting parties; translated as to accomplish something by interposing between two parties, to mediate, (with the accusative of the result): τήν διαλυσιν, Polybius 11, 34, 3; τάς συνθήκας, Diodorus 19, 71; Dionysius Halicarnassus 9, 59; (cf. Philo de plant. Noë, 2:2 at the end).

2. as a μεσίτης is a sponsor or surety (Josephus, Antiquities 4, 6, 7 ταῦτα ὀμνυντες ἔλεγον καί τόν Θεόν μεσιτην ὧν ὑπισχνουντο ποιούμενοι (cf. Philo de spec. legg. 3:7 ἀοράτῳ δέ πράγματι πάντως ἀόρατος μεσιτευει Θεός etc.)), so μεσιτεύω comes to signify to pledge oneself, give surety: ὅρκῳ, Heb 6:17.