The prefix "ha" is, of course, the definite article ("the") in Hebrew words.
Most people would translate "gibor" as hero. While we would probably acquiesce to the suggestion that God is indeed the hero of Jewish history (and even the world), it seems a little too human, even as a metaphor, to suggest that our image of God compares to the individual who responds to crisis, rather than one who is constantly present in life.
Still, the hero is the one who knows the right thing to do in any given situation, and that certainly describes how we imagine God. But what is heroic is not always that which demonstrates another common translation of gibor: strength.
Instead, I suggest a better meaning for gibor in this context is "courageous." Courage is the wisdom and strength to do the right thing, not just to do the powerful thing. I like to think that God's wisdom enables God to do what is right, even when it is not obvious or popular. The familiar adage in the Talmudic tractate "Avot" says, "Who is courageous (gibor)? The one who controls his impulse." It is certainly human impulse to enforce our own needs and desires. It takes enormous self-awareness and self-control not to exercise our strengths, physical or otherwise, when we feel our just values are violated.
There is, therefore, a continuing courage on God's part in controlling the divine impulse to punish disobedience or to overrule nature when the result is undesirable. How ironic that what we plead for – God's miraculous intervention – might strengthen our faith, but foil God's integrity. To know the consequence of self-control but exercise it anyway – that is indeed courage.
There is also a Modern Hebrew meaning to "gibor" that strikes me as apropos. In a story, the gibor is the central character, much the way English speakers refer to that character as "the hero of our story." Though unintended by the authors of the prayer, this meaning may be the most appropriate of all. The characters who occupy a central place in Biblical stories come and go. Once their actions or their lives have come to an end, so does their centrality. But it is accurate to say that God remains the central character through the Bible and beyond into even moment of history.
When we pray using this word to reflect one of God's attributes, we acknowledge that God is central to the story of the Jewish people and, we hope, to our own lives.