The Euthyphro dilemma, though designed for a more polytheistic worldview, is occasionally used against Christianity. Does God love good because it is good, or is good good because God loves it? The argument suggests that if God loves good because it is good, there is a higher standard of good to which even God must be held accountable, and if good is good because God loves it, God is too arbitrary, and we're no better off than if a human being was writing the rule book.
But the truth is that with God, it doesn't need to be an either or. There are many attributes God has that neither human beings, nor the gods we imagine, possess. Most relevant to this "dilemma", God is Holy. This means that God Himself is the highest standard. And He cannot change, and is therefore not arbitrary.
Let's compare this to a human design. Does an artist love his work because it is good, or is it good because it is loved by the artist? In this scenario, the "goodness" of the art is entirely subjective. Only the artist can really tell you whether or not he achieved his goals. You might look at it and see a "flaw", and yet that "flaw" might have been entirely intentional. If the artist achieved his goal, the art is objectively good, and so the artist will love it.
In the same way, good is anything that achieves God's desires. This isn't arbitrary at all! While unbelievers act as if God can simply say "murder is totally cool now", murder will never be good, because murder is the opposite of God's desire. Thus, life is good because it is loved by God, and God loves it because it is good, and there is neither contradiction nor inconsistency in that.
This is where the unbeliever might reply by pointing out that God is perfectly capable of taking life, and aside from the fact He is the ultimate cause of all death, He has directly killed, or commanded the killing, of many people. After all, if I'm saying "God is Holy, therefore murder is always wrong", that is contradictory, and proves God is evil by His own standards. Right?
It is entirely understandable that one might think that way, but there are other factors to consider. The first of these is "property rights". Think of it this way: You could pick the lock of your own front door in front of as many police officers as you want, there is nothing they could do about it. It's your property. Similarly, if you permit someone to pick that lock, and especially if you command someone to do it for you, like a locksmith, there is no crime committed. But if someone picks your lock without your consent, that's a crime.
The second thing to account for is that it has never been as simple as life good, death bad. Death is actually a course correction, both as a method of directly punishing sin, and indirectly redeeming that which is stained by sin. God, of course, tends towards life. As sinners, we are literally living proof that God prefers life. The worst of sinners, while they still live, has a chance to repent, and receive the gift of life so freely offered by, of all things, the death of the only Son of God. The fact that Jesus, who never sinned, and thus never earned death, nevertheless died that the guilty may live, shows that God tends towards life even when death is a justified option. A time will come when this entire creation will be dead, the very elements being burned up by a fervent heat, yet those who put their faith in Christ will live. And of course, scripture is abundantly clear that this is God's preference.
In other words, God is literally incapable of murder. He has the right to give and take any life He sees fit, and He sees fit far more often than we have any business questioning Him about. His Holiness is shown in His excess. He is just, and so there will be those who suffer the second death. Yet, eternal life is still on offer.
From here, we can actually flip the Euthyphro dilemma, even to the point of using it as evidence for God. See, if there is anything good at all, there must be a God on top of it. Even if we were to suggest there is a higher power governing Him, that higher power would then be God's god, and we would have to apply the Euthyphro dilemma to that god.
So where do we go from here? Well, we could tend towards total anarchy. Maybe there is no good and evil. But unless you're a literal psychopath, you instinctively know that's not true. So we know that good and evil do exist. The question we must ask is which god governs it? The obvious answer is not us. We change like underwear, we differ like a tiger's stripes, we err like the humans we are. Where would we even get our authority, if we have the audacity to claim it?
But to go higher than God is absurd. As He Himself asks, "Who hath prevented me, that I should repay him? whatsoever is under the whole heaven is mine." (Job 41:11). God's authority comes from His Godliness. For lack of better term, there was a "time" when it was just God. Father, Son, Holy Spirit, living in perfect fellowship in eternity. What governed Him then, but Himself? Nothing. And when He created everything, did He include an authority for Him to submit to? By no means! The absence of external governing force persists to this day.
The one and only time God has been governed by a higher power is when the Son entered the creation, submitting Himself entirely to the Father. And He succeeded perfectly. When scripture commands us to be Holy as God is Holy, we failed, but the Son of God made flesh succeeded. Yet He died, that we, through faith, might live.
The Euthyphro dilemma, therefore, fails against God. Rather than show an inconsistency in the faith, it presents us with both great evidence that He exists, for moral laws require a moral lawgiver, and shows us our need of Him, for we have broken His laws. Let us therefore come to God on His terms: Humbled, penitent, faithful.