The prophecy in Isaiah 9 seems primarily concerned with the survival of the throne of David in the kingdom of Judah (verse 9.7,21) under the threat of Syria and Israel (9.9,11-12,21). This fits the historic context of the eighth century BC, as well as the context of the immediately preceding chapters, Isaiah 7-8, which is concerned with Judah in relation to the kingdoms of Israel, Syria, and Assyria.
Broadening our context from Isaiah 9 over to chapters 7 and 8, we find three examples of children being named according to their respective occasions. If we broaden our context to Isaiah's contemporaries, we find another prophet also giving names to children in Hosea 1.
- She'ar-yashub = 'A remnant will return' (Isaiah 7.3), but no context is given
- 'Immanu-'el = 'God is with us' (Isaiah 7.14), as a sign that God will protect King Ahaz's kingdom from Syria and Israel by sending Assyria
- Maher-shalal-hash-baz = 'Quick to the spoils, swift to the plunder' (Isaiah 8.3), descriptive of Assyria's conquest of Syria and Israel
- Yizre'e'l = named for valley of Jezreel (Hosea 1.4), descriptive of God's divine vengeance upon 'the house of Jehu', apparently for killing Joram in the valley of Jezreel (e.g. 2 Kings 9.14-29)
- Lo-ruchamah = 'not pitied' (Hosea 1.6), descriptive of God's lack of pity for Israel in its time of punishment
- Lo-'ammiy = 'not my people' (Hosea 1.9), descriptive of God's disowning of Israel
The name is not intended to describe the child. Rather, the names describe the prophetic context to which each child was born. It seems probable that Pele'-yo'ez-'el-gibbor-'abiy'ar-sar-shalom, if it is indeed a prophetic name is not meant to describe the child, but the situation to which that child is born (established in Isaiah 7-8): that God would preserve the throne of David in the kingdom of Judah through the threat of Israel, Syria, and Assyria.
This seems verified by the phrase found in the next verse, 9.7: 'The zeal of YHWH of hosts will do this'.
While Isaiah did not, of course, write the historical appendix found in Isaiah 36-39 (= 2 Kings 18.13-20.19), these chapters are part of the final form of the book, and shed light on how Isaiah 9.6-7 was understood in the sixth century BC by the book's editors.
That particular phrase, 'The zeal of YHWH of hosts will do this', is not found anywhere else in the Hebrew scriptures, nor the New Testament... except for Isaiah 37.32 (= 2 Kings 19.31), where it is found in a historical context immediately relevant to Isaiah 7-8: Judah is under threat by Assyria, after Assyria had just conquered Syria and Israel (2 Kings 16.1-18.12, which is just prior to the section that has been copied from 2 Kings into the book of Isaiah).
Who the Name Refers To
Given all of the above, it seems most likely that the prophetic name Pele'-yo'ez-'el-gibbor-'abiy'ar-sar-shalom refers to King Hezekiah, and describes what God would do for Judah through that king's reign.