A post at the Secular Outpost pointed me to an old post of Victor Reppert's, where he summarizes five arguments in favor of moral objectivity. I want to consider these arguments briefly.
First, though, I think Reppert has confused the subjective-objective issue with the relative-absolute issue. This SEP article explains the difference, but let's review it here:
If morality is objective, it is something that exists "out there in the world", independent of what anyone thinks or believes. The truth of a moral claim is fixed by objective facts.
Moral subjectivism, on the other hand, says that moral truth is fixed by some person or persons. The person could be the individual, the social group, or God. (So William Lane Craig is being inconsistent when he argues that morality is objective, yet subscribes to a Divine Command theory of morality: the latter ascribes morality to the desires of God, and thus it depends on a person's (God's) opinion. That's subjective, not objective.)
The other axis is relativism versus absolutism.
Morality is absolute if it is the same for everyone, everywhere, at every time.
Morality is relative if it depends on the person or the social context. As the SEP article says, "Stealing is wrong" could be true for one person and false for someone else, for instance, for someone from a different culture where stealing is an acceptable practice.
These axes are "orthogonal", in the SEP's words: it is possible for a moral theory to be subjective-relative, subjective-absolute, objective-relative, or objective-absolute.