Is there a way to believe that God is the source of morality without also having to believe that morality is arbitrary?
The most handy solution that I know of is provided by William Lane Craig:
...does God will something because it is good, or is something good because God wills it? If the theist says that God wills something because it is good then the good is independent of God and, in fact then, moral values are not based in God. They are independent of him. On the other hand, if you say something is good because God wills it then that would seem to make what is good and evil arbitrary. God could have willed that hatred is good; then we would be morally obligated to hate one another, which seems crazy. Some moral values seem to be necessary, and therefore there would be no possible world in which hatred is good. So the claim is that this shows that morality cannot be based in God.
I think it is clearly a false dilemma because the alternatives are not of the form “A or not-A” which would be an inescapable dilemma. The alternatives are like “A or B.” In that case you can always add a third one, C, and escape the horns of the dilemma. I think in this case there is a third alternative which is to say that God wills something because he is good. That is to say, God himself is the paradigm of goodness, and his will reflects his character. God is by nature loving, kind, fair, impartial, generous, and so forth. Therefore, he could not have willed that, for example, hatred be good. That would be to contradict his very own nature.
So God's commands to us are not arbitrary, but neither are they based upon something independent of God. Rather, God himself is the paradigm of goodness.
If we follow the Neo-Platonic point of view, then the Euthyphro has an implicit solution, which is that God is the Good itself. If God is simultaneously the source and the measure of all goodness, the paradox disappears. This is arguably why the paradox is outlined in the Euthyphro in the first place, as a Socratic prompt to that conclusion.
It's worth noting, however, that this solution works better for an abstract, non-personified conception of God than for God as envisioned with concrete, specified traits. One of the charges against Socrates, as you may remember, is that he weakened the belief in the traditional Greek Pantheon of quarreling, anthropomorphic gods.
Here's another suggestion, maybe not all together different than the William Lane Craig quote from elliot svensson.
God's nature grounds the truth of moral propositions (his nature is their "truthmaker", or that in virtue of which they are true) - they are therefore not independent of God, but at the same time they are not arbitrarily willed by God since they are absolutely true because of some objective and essential fact about him that God recognizes to be true.
So, if God commands us to refrain from acts of gratuitous harm, for example, it is because it is true that we ought not to commit such acts, and it is true because of some essential fact about his nature. There is no appeal to anything outside of God. Morality depends every bit on God as God's omnipotence or some other essential attribute of him. If God did not exist, neither would God's omnipotence, and neither would that which grounds the truth of moral propositions (and so there would be no true moral propositions). But it's not arbitrary, as God couldn't choose to not be the ground of moral propositions anymore than he could choose to be not omnipotent, or make 1 + 1 = 3.