Having been begotten again (αναγεγεννημενο). Perfect passive participle of αναγενναω, which see in verse 1 Peter 1:2.

Not of corruptible seed (ουκ εκ σπορας φθαρτης). Ablative with εκ as the source, for φθαρτος see verse 1 Peter 1:18, and σπορας (from σπειρω to sow), old word (sowing, seed) here only in N.T., though σπορος in Mark 4:26, etc. For "incorruptible" (αφθαρτου) see verse 1 Peter 1:4; 1 Peter 3:4.

Through the word of God (δια λογου θεου). See James 1:18 for "by the word of truth," verse 1 Peter 1:25 here, and Peter's use of λογος in Acts 10:36. It is the gospel message.

Which liveth and abideth (ζωντος κα μενοντος). These present active participles (from ζαω and μενω) can be taken with θεου (God) or with λογου (word). In verse 1 Peter 1:25 μενε is used with ρημα (word). Still in Daniel 6:26 both μενων and ζων are used with θεος. Either construction makes sense here.