I would say that Alvin Plantinga's free will defense is perhaps the most philosophically rigorous defense that many agree defends the logical possibility of a good God which has created a world with evil. Please see the link for arguments pro/con about his defense.
Let's be clear about the argument - it's not just that God always acts good (or is all-good/omnibenevolent), it is also that God is all-powerful and all-omniscient, but evil still exists - how can a good God create/allow moral evil?
Platinga's argument is a logical attack on the core ability of an all-powerful being to logically create a world in which a true moral actor can always act good. However, I think that leaves the argument rather dry and makes it easy to miss the deeper logic of it.
There Must Be A Greater Good
The core of the defense must be that there is a "greater good" in allowing free moral choice and thus permitting - and even creating - evil in order to preserve free moral choice, because that is the morally right thing for God to do. It is morally right for God to give humans free will, even though a side result of giving free will is that they are able to commit evil.
Good must be defined as "whatever God wills", because God is defined in the "Problem of Evil" argument as all-good. Therefore evil cannot be committed by God - again, he has been defined in the argument to only do good. Evil must be defined as being committed only when a free moral actor other than God chooses to do something other than what God wills. Therefore it is never God desiring or willing evil, but only permitting the evil that other moral actors commit, because - and this is the key - to not permit a free moral actor from committing evil would be to override free will, and free will must therefore be a greater moral good than permitting evil.
I do believe this answers the problem of evil, per your question.
Why Is Free Will a Greater Good?
Some will ask why allowing free moral will is a "greater good" which overcomes the fact that evil must be allowed in this world. I believe this can be answered as follows:
1) God is love, he made us to be in relationship with him and with each other, to love and be loved. This is evident throughout the entire metanarrative of the Bible from a Christian perspective. For example, Jesus said, paraphrased: "Love God above all, and your neighbor as yourself, these commands sum up all of the law". The ability to love is therefore the essence of all good moral conduct.
2) Given #1, the argument is that the greater good for allowing evil is to allow love. Therefore, an assumption made is that free will is required in order to allow love. If one cannot choose to love, one cannot love. The definition of love must require free choice to love - if one is compelled to love another being it cannot properly be called love. A world without free will is therefore a world without the greatest moral choice - love.
3) Given #1 and #2, it follows that for God to create a world where the highest moral good is possible, he must create a world with love. To create a world with love, he must create free moral actors. Free moral actors must be allowed to choose either good or evil by definition. Therefore, evil must be allowed in order to allow free will and thereby allow the greater good of love to exist.
A good God therefore must create a world in which free moral actors other than God himself are allowed to commit evil moral acts, because that is the only way that the greater moral act of love can also exist.
Put another way - to create a world without evil is to create a world without love.
Note: Free moral will in this defense is Libertarian Free Will, not Compatibilist Free Will
Edit to add comment regarding last 2 paragraphs:
1) A world without Hitler would thus be a world in which no human is able to love each other or have a relationship with God, greater than perhaps a dog might have with a man. Is a world of "not-evil" robots better than a messy, but meaningful world of real humans with real relationships and choices? I say no, but your mileage my vary!
2) It is wrong to say heaven is the best possible world - your view here is too narrow - you must include all of creation/the universe/the multiverse in order to make this argument. Thus a universe including both existing earth and heaven, future earth and heaven (if different than current), and also hell, must be considered. If you boil the defense above down to allowing relationship with God and choosing either for his way or against it, then heaven just ends up being the place where you continue that relationship, and hell is the place where you go to continue not having a relationship with God, so they are sort of just an extension of your choices in that regard here on earth and morally the spiritual world is not then much different in analysis from the material world.