You've probably heard before that forgiveness is a gift you give yourself. That means that in our acts of forgiveness we reward ourselves. The implication is that the motivation for forgiveness is not to relieve the suffering of the forgiven person but that of the forgiver.
Despite having heard the above concept of forgiveness, we still generally struggle with forgiveness, largely, I believe, because we expect people who have wronged us to be punished. Somehow we have been convinced that by not forgiving, we make the wrong doer suffer more. Our unwillingness to forgive is motivated by a desire for revenge, and anything that diminishes our sense of justice is rejected.
Jesus noted the human reluctance to forgive in the Sermon on the Mount when he said we would experience forgiveness to the same extent that we forgive others. Though the words of Jesus make God's blessing of forgiveness contingent upon our human capability to forgive, I suspect the true meaning is that our capacity to enjoy the blessings of abundant life which God intends, symbolized by the term forgiveness, is contingent on our exercise of forgiveness. If we miss being forgiven or blessed with abundant life, it will be because we failed to love and forgive as God does. That missed blessing is not God's retribution but rather a natural consequence of being created in God's image.
Theologically, Christianity teaches that God's forgiveness is granted for our benefit. If God decides to forgive, we escape His punishment. God either forgives, or He punishes the unforgiven ones. God is satisfied either way, apparently. He is perfectly willing to forgive or punish. Supposedly, His decision to forgive is prompted by man's actions in soliciting His forgiveness. In effect, each individual controls God's decision about forgiveness. God does not decide on His own to forgive, each individual decides personally to activate God's forgiveness. Once the man decides, God's forgiveness is assured. God must forgive because that is the promised result of deciding to activate God's forgiveness appropriately.
The workings of divine forgiveness raise the question of whether God benefits when He forgives. Christian Orthodoxy would lead us to believe that God benefits in forgiving a few by gaining faithful followers who will properly honor and obey Him throughout eternity. His forgiveness is a benefit because it is the process by which God culls humanity for the chosen few who will finally fulfill the purpose for which mankind was created in the first place, to serve His pleasure. Of course, by extension the lack of forgiveness for the many, those who suffer eternal punishment, must also benefit God in some way, else God is forced to do something detrimental to His own well being by some power greater than He. That hardly seems god-like, so eternal punishment must be divinely beneficial.
Some may want to suggest that the greater power which constrains God is His very divine nature, which is inviolable even If God would will it to be violated. This is the old standard argument that God cannot overlook sin; He is just too holy. What we see in these rationalizations is an attempt to overrule one aspect of God's nature by another more powerful one. God loves and wants to forgive, but He is trumped by a greater need for justice and obedience. It makes for a confused picture of a convoluted god.
So does God benefit from forgiving us? Perhaps not in the same way we humans benefit from forgiving others. When God forgives, He simply expresses His true nature, serving as an example to mankind. When we forgive, we momentarily embrace a divine nature which more often than not eludes us. God's forgiveness reflects who He is, and ours reflects who we potentially become.