Option 1 is almost certainly what Isaiah meant
El-Gibbor strongly parallels names like Ishmael ("God has hearkened") and Elizabeth ("God's promise"). According to a footnote in the NET Bible:
גִּבּוֹר (gibbor) is probably an attributive adjective (“mighty God”), though one might translate “God is a warrior” or “God is mighty.” Scholars have interpreted this title is two ways. A number of them have argued that the title portrays the king as God’s representative on the battlefield, whom God empowers in a supernatural way (see J. H. Hayes and S. A. Irvine, Isaiah, 181-82). They contend that this sense seems more likely in the original context of the prophecy. They would suggest that having read the NT, we might in retrospect interpret this title as indicating the coming king’s deity, but it is unlikely that Isaiah or his audience would have understood the title in such a bold way. Ps 45:6 addresses the Davidic king as “God” because he ruled and fought as God’s representative on earth. Ancient Near Eastern art and literature picture gods training kings for battle, bestowing special weapons, and intervening in battle. According to Egyptian propaganda, the Hittites described Rameses II as follows: “No man is he who is among us, It is Seth great-of-strength, Baal in person; Not deeds of man are these his doings, They are of one who is unique” (See Miriam Lichtheim, Ancient Egyptian Literature, 2:67). According to proponents of this view, Isa 9:6 probably envisions a similar kind of response when friends and foes alike look at the Davidic king in full battle regalia. When the king’s enemies oppose him on the battlefield, they are, as it were, fighting against God himself.
Given the context of the passage, which envisions a future kingdom ruled by a descendant of David who will defeat Israel's enemies and establish lasting peace, it makes sense that God's instrument will bear a name honoring God's military power. The other names also seem to refer to God and are born by the coming king as a reminder that: "The zeal of the Lord of hosts will do this." (Isaiah 9:7b ESV)
Option 2 works, but is unlikely
The NET Bible footnote continues:
The other option is to regard this title as a reference to God, confronting Isaiah’s readers with the divinity of this promised “child.” The use of this same title that clearly refers to God in a later passage (Isa 10:21) supports this interpretation. Other passages depict Yahweh as the great God and great warrior (Deut 10:17; Jer. 32:18). Although this connection of a child who is born with deity is unparalleled in any earlier biblical texts, Isaiah’s use of this title to make this connection represents Isaiah’s attempt (at God’s behest) to advance Israel in their understanding of the ideal Davidic king for whom they long.
It would seem that nothing grammatically excludes the interpretation of name from referring to God, since it's used in Isaiah 10. But the suggestion that Isaiah is expanding "their understanding of the ideal Davidic king" is out of step with the culture he is writing from. More damaging is that the next title, "Eternal Father", makes little sense in the Trinitarian framework. (The NET Bible note on that phrase is helpful too.)
For those scholars who are committed to a Christian view, the phrase can be applied to both to the Davidic king of Isaiah's age and also to Jesus Himself. This would be an example of double fulfillment. But this interpretation must be limited to those who already accept the framework of Christian theology. In isolation, Isaiah does not support this interpretation.