If picture quality is paramount, forget HDMI over ethernet. A 1080p60 image takes about 4.45 Gbps. It will have to be compressed to send over ethernet. The best products claim to be "visually lossless" but they are still lossy. Best bet is an HDMI extender based on HDBaseT that runs over a single Cat 5e/6 cable. They are truly lossless.
Does anyone have some good recommendations? I see some on google for as little as 16 bucks.... and others for over 300.... with no real difference that I can tell.
Again, the Monoprice HDBaseT model... $180.
Your run is relatively short, and if you're going to run wire anyway, you can use one of the ~$50-75, two-cable solutions. Steer clear of any of the "unpowered" solutions (those without a DC power adapter). There will be feature differences as well, whether they offer IR repeating, is usually the big one.
There is a thread dedicated to this topic. The key thing is to use an extender that has its own power supply. The ones that use the 5V line on the HDMI cable for power generally don't work well. The ones that use a single cable generally seem to work better than the ones that use two cables. HDBaseT is the way to go if you don't want to waste time finding something that works for your situation.