Maybe the speed of light was faster in the past? (Quite a far-fetched suggestion given that the difference would have to be almost 6 orders of magnitude, which is just humongous.) Maybe the universe is smaller than we think? Maybe God just put the light on the way? Maybe time for us is different from the time in outer space?
This article in answers in genesis summarizes quite well all the pseudoscientific handwaving that creationists have come up with trying to explain away the contradiction.
How to answer such pseudoscience with a relatively easy to understand refutation? Let me present to you SN 1987A.
SN 1987A was a supernova that was first observed in 1987. It was a very peculiar supernova in that the star was surrounded at a great distance by a visible dust cloud (which was created by the ejected material of another supernova in a very distant past). This dust cloud was illuminated by the supernova several months after the explosion. Since we know the propagation speed of light, and we could measure precisely the time it took for the light of the supernova to reach the dust cloud, we therefore can calculate how far the dust cloud is from the supernova and by triangulation how far it is from us. It turns out that the supernova and the dust cloud are approximately 168 thousand light-years from us.
This means that the star exploded 168 thousand years ago (and illuminated the dust cloud several months later). This is an active event that happened back then. It's not just a star sitting around idly for 6 thousand years and God putting light from it on the way.
If the explanation that the speed of light was faster in the past, which would mean that it would have to had been about six orders of magnitude faster to account for the farthest galaxies being visible, that would mean that the supernova was actually six orders of magnitude farther away, at over 100 billion light-years. A supernova that far away would most certainly not be visible to us. (Heck, even a quasar would probably no be visible to us from that distance.) Not to talk about the size of the dust cloud... (I think it would be way larger than the largest galaxies.)
We being in a deep gravity well and hence our time traversing at a different rate than the time in the space between galaxies is bollocks (well, not totally of course, because we are in a gravity well; however, it's not that deep), but even if it weren't, it doesn't matter: The star exploded 168 thousand years ago from our perspective. Those are the exact same years as the 6 thousand that are alleged as the age of the universe. (The speed of light is constant regardless of who is measuring it. It doesn't matter if you are in motion, accelerating or in a deep gravity well, the speed of light will always be the same. That's the very foundation of general relativity and has been corroborated to death. Thus light couldn't have traveled faster from any perspective at a distant location from us. It took it 168 thousand of our years to reach us.)
By the way, this conclusion in the answers in genesis article above made me literally laugh out loud:
Answers in genesis wrote:That is why, ultimately, the only way to know about the past for certain is to have a reliable historic record written by an eyewitness. That is exactly what we have in the Bible.