Most short names in Greek are based on the first member of a compound (e.g. Ζεῦξις : Ζεύξ-ιππος and Εὐρυσθεύς : Εὐρυ-σθένης).
Some are based on the second member, such as Δίκα (Sapph.81.4, vocative) : Μνᾱσι-δίκα (Sapph.82a, nominative or vocative) and Φράδμων : Πολυ-φράδμων (Πολυ-φράσμων); Αἵμων : *Εὐ-αίμων and *Πεδώ : Ἐμπεδώ. These examples are from Schwyzer, GG, 636.
So, we have short names of the Eliza or Chris or Alex type and of the Beth or Topher (or Pher) or Xander (or Sandy) type.
Of a later date, but still instructive is Ντίνος (beside Κώστας) from Κωνσταντίνος. See A. A. Thompson 2007: 684.
The same set of examples in Schwyzer includes 'Monkey-Boy', Πίτθων (cf. Πιτθεύς Il.3.144: Πίθηκος (see Schwyzer, DGE no. 416.15 and note, and the Arcadian inscription cited by LGPN).