The KJV and modern versions differ in this verse:

And he said, Thy name shall be called no more Jacob, but Israel: for as a prince hast thou power with God and with men, and hast prevailed. (KJV)

He said, “Your name shall no longer be Jacob, but Israel; for you have striven with God and with men and have prevailed.” (NASB)

Scholars today do not think that the KJV is correct (e.g. Westermann). The Hebrew (MT) does not have a comparative noun-phrase ‘as a prince’ and the Hebrew verb underlying ‘as a prince hast thou power’ (KJV) is unique to this text and Hos 12:4 which is a commentary on Genesis. The meaning of the verb is being taken from the incident, hence the NASB has ‘to strive’ but versions are free to pick and choose something similar. Two of the consonants from the verb’s root (hrf, larfy) are incorporated into the Hebrew of ‘Israel’ along with the title of God—’ēl. They are also the two consonants in the Hebrew noun for ‘prince’ which may explain the KJV translation.

The emphasis in Hosea is on God who was manifested in the angel. Jacob has this understanding that he is wrestling with God in the angel because he names the place Peniel (‘the face of God’):

He took his brother by the heel in the womb, and by his strength he had power with God: Yea, he had power over the angel, and prevailed… Hos 12:3-4 (KJV)

The KJV drops ‘as a prince’ here and Hosea gives us another verb in ‘power over the angel’ but it is also rare (rwf, 3x). It doesn’t necessarily help us fix the meaning of hrf, although there may be a word-play in its use. Translations vary in their rendering of the verb rwf in Hosea (RSV, ‘to strive’, NASB ‘to wrestle’) and the few other occurrences don’t help. Again, it seems the incident is driving translational choices.

The name ‘Israel’ means whatever the translation indicates: for example, something like ‘God will have power’ (KJV) or ‘God will strive’ (RSV). However, the use of the common verb for ‘to prevail/to be able’ (lky) to sum up Jacob’s success is not incorporated into the name ‘Israel’.

Genesis also mentions ‘and with men’. Why is this? Hosea mentions Jacob taking Esau by the heel which would be the beginning of Jacob’s struggle with men. But we don’t have to restrict the ‘and with men’ just to Esau since Jacob’s life saw him struggle with men. Is the ‘and with men’ important to the name ‘Israel’? It is attractive to say so because ‘God will have power with men’ fits the story of salvation-history. Perhaps we are meant to infer that God will strive with men from use of the name. Hebrews 2:14 is relevant here because it quotes ‘took his brother’ in the phrasing ‘took part of the same…the seed of Abraham’. This would make Jacob’s struggle with men a type of God’s struggle. Giving the name ‘Israel’ to Jacob in effect declares that God will strive with men ‘in Jacob’ which is a prophecy of Christ.

Striving with God is obviously a different function than striving with men. Insofar as the name ‘Israel’ incorporates a title for God with ‘to have power’ (KJV), it is excluding the ‘with God’ element: God does not strive with God. This shows that it is the ‘striving’ that is important to the name. It is the angel that shows that God will strive with men.

Hosea continues,

Yea…he wept, and made supplication unto him: he found him in Bethel, and there he spake with us… Hos 12:4 (KJV)

What is interesting is that here we do not have further description of the wrestling. Rather, the statement is summarising aspects of Jacob’s life. We can tell this because weeping and making supplication are not part of the wrestling but the same verbs are used together in Jacob’s meeting with Esau (Gen 33:4-5); and finding God in Bethel is recounting Jacob’s encounter with God in Bethel where it is also emphasized that he ‘spoke’ with him (Gen 28:16; 35:15).