The root *ntn in Hebrew denotes giving and the verb in the perfect aspect natan, is either in the past tense or something eternal.
Here of course is also the meaning of the name Nathan – ‘he gives/he has given’. This root also is at the center of the name Nathanael. Nathanael is in meaning almost identical to Jonathan as it means ‘God gives/God has given’. The yo in Jonathan referencing the LORD (yhwh) and the el in Nathanael referencing God (elohim).
The grammar of Jonathan also allows for the translation ‘the LORD he has given’, the object being the LORD. This is possible since finite verbs in Hebrew inherently has a subject with person and number. So, in the name Jonathan, yo can taken to be either an explicit subject (the one who gives) “The LORD has given” or the object (the one who is given) “He has given the LORD“…
Whether this actually has any bearing on Christology is a different question – certain however is that the noun matanah “gift” (cf. Gen. 25:6) is from the same root *ntn and that Jesus the Lord is God’s gift to the world:
“For God so loved the world that he gave his only Son, that whoever believes in him should not perish but have eternal life.” (John 3:16) Indeed, since the Father and the Son are One (10:30), Jesus can say: “I give them eternal life, and they will never perish, and no one will snatch them out of my hand”. (John 10:28)