I only have a very old apple TV, so I am going to pass you the baton to discover how to do some things.
You can find out which one you have from the model number on the base.. just look at the A1xxx number. You might need a magnifying glass..
Gen 2 TV is A1378 .. A Gen3 is A1427 or A1469
This is the stuff you just google. Identifying Apple TV models - Apple Support
Edit.. just saw Bob's suggestion.. of course it should show the model on the screen in big letters.. no magnifying glass required.
What happens with Yosemite is ghosting.. so it keeps dropping out and when it comes back it changes its name. So you will see Yosemite also has (xx) number after the name. In my experiment with Yosemite not only was yosemite computer affected but other devices on the network were also forced to change name.. that may have been unusual.. not many people are posting such issues.. but perhaps it was good for me to see it. as that is why I think the problem with your apple tv is caused by Yosemite.. especially if it never happened until Yosemite was introduced. Can you confirm when did this start happening... when Yosemite arrived??
1. When you mention reset to factory default and locking the name I assume that you are referring to the Apple TV?
Yes, I do want you to factory default the Apple TV but locking its name is probably not possible. I wasn't being as clear about the capability of ATV as i should have been. However what I was intending was for you to set its IP statically. I am not sure of the naming and how much control you have.. but let me encourage you now that you have Yosemite in your midst to keep all network names to cross-platform standards.
OS X: Cross-platform filename best practices and conventions - Apple Support
In summary.. and slightly perverting Apple's instructions.. from long and bitter experience with windows.. Keep all network names short, no spaces and pure alphanumeric. Keep passwords the same.. mixed case and numbers.. but longer 8-20 characters.. Keep things simple. Apple will always recommend names with spaces and apostrophe.. do not accept them.
But instead of leaving your apple TV to dhcp.. I want you to set it statically on the device and see if that fixes the problem.
Or you can lock it via dhcp reservations in the AE. (assuming it is in router mode).
2. When you say to use ethernet for Apple TV would that entail running an ethernet cable from the ATV to the Airport Extreme?
Yes, but again this is a test to solve a problem.. if it works there are alternatives.
Do this if # 1. fails.. move the ATV close to the AE or AE close to the ATV.. whatever works for you.. and use a short patch lead.. there is no need to do major room changes.. plug the apple tv into a standard computer monitor or small TV and see how it goes.. I am thinking the problem is more likely just wireless side. If it fixes things again you can use EOP adapters rather than having to run ugly cables. (blue ethernet cables over your white carpets.!!)
You are looking for the source of the issue.. and trying to see if simple things like using wired instead of wireless connections fix it.
EDIT..
Tell me your whole network layout.. I would like you to try a simple change of dhcp lease.. and see if that might fix it.
All you need is to fix the network stability.. and to do that we need to find out why it has become unstable.