Finally, Whitespike gets a turn with the inevitable failed shock mount  

Hume
I went the Hume route. Honestly it was worth the 35 bucks or whatever I paid. Sent it off, got it back. No mess, no fuss. Chair is good as new.
As always its tricky to determine wear - I'm not sure the fault lies more with the chairs, or how people use the chairs. Under normal use I can imagine the mounts popping off once in a while.
But 7 times in ten years? That seems like someone is either abusing the chairs, or using the wrong adhesives.

Alfie's estimate for the repa...
Alfie's estimate for the repair was $125. I'm not paying over $100 every time one of these fail... I'm going to go the epoxy route methinks.

Use the right adhesive
We've had extensive discussion about the best adhesive for shock mounts, and it comes down to two:
1) Scotchweld 1838 - that's the 3M product originally used by HM
or
2) PC-7 - which seems to be much cheaper and easier to find.
I have chairs repaired with the Scotchweld 15 years ago that are still holding up. The stuff in unbelievably good (if overpriced).
You can handle this, Whitespike.

Seriously?
His prices have gone up. I swear mine was less than $50 including shipping both ways.

As Koen
has pointed out in the past, I believe, you don't want to squeeze the joint so much that there is no adhesive left in some spots. The bag-weight solution is good, with the chair-back taking some of the weight and (not incidentally) keeping the weight and the puck in place. An initial piece of tape would be good, too, to keep the puck where you want it.
I would think that careful removal of existing glue and torn rubber would be important; you want to glue solid rubber to solid wood, with nothing in between.

If I removed the thin layer...
If I removed the thin layer of rubber, then the existing mount wouldn't be stable. I have planned on using the original mount. Is this not sound?

Right --
In your case, part of the rubber remains on the chair back. I should not have suggest that you remove that., if you plan to reattach the existing puck. But anything loose needs to have glue under it as you proceed.
I don't know what the experts would counsel: whether to remove the remnants, discard the damaged mount, and start with a new one, or to proceed with what you have . . .

The scotchweld can be found....
The scotchweld can be found. At about 45-50 bucks including shipping for 2 oz. The PC-7 can be had a Home Depot for around 12 bucks. I don't mind paying 50 for a much better product. But honestly I am eager and would like to just go get it and get started this weekend.
Any last words before I go buy the PC-7? Should I hold my horses and go for the Scotchweld?
I think I'll use a large bag of rice to stabilize the back and use another bag on top to even out the surface. I'll use painter's tape to mask the wood and regular tape to hold the mount in place. Lastly I think I will use heavy books on top.
I wish they made PC-7 in clear.

mouse trap
If you search the term "mouse trap" on this forum or "shockmount" it will lead to some prior discussion.
I mentioned there and will say here that I used 3M Scotchweld DP-190. It is a clear product. The 1838 has been mentioned also as a good option. Special K discusses some testing they did and concluded that the PC-7 is easy to work with and easy to purchase. The Scotchweld is tougher to find. I found it in the Minneapolis area through Fastenal. It has an expiration date of about 1 year from manufacture, but I believe poach said he has used the same twin tube for 10 years, so.....
I made a clamp system, but was glueing a lozenge shockmount on an LCW and needed to have it all on the chair frame to hold its curvature. A puck probably doesn't matter as much.
If you find the weight system difficult. You can make a big sandwich clamp using threaded steel rods, 2 pieces of wood, wing nuts and packing foam to protect your wood. It will give you nice even pressure. This worked great for me!

One word...
Teenagers!
To be honest, I contacted HM and was told to send in the entire chair for evaluation. Then I contacted DWR, from whom I purchased the chairs, and they admitted that they'd had many returns from that run of DCM's and that they suspected a bad batch of adhesive was to blame. Anyway, sending the entire chair seemed way more trouble than making the repairs myself and, so far, I've had a zero failure rate and have become quite adept at it to boot!
PC-7 works great. You don't need a lot. I like to remove all of the residue from both parts first. This can be some work. I use small palm gouges and chisels that I have for carving wood blocks for printing, but a rotary tool like a dremel would be faster. Either way, this step requires the most caution. Roughen the back of the puck with course sandpaper to provide more "tooth". Clean all surfaces to be glued with a Q-tip and acetone trying not to get any on the wood finish. Mix PC-7 well and spread a very thin layer completely over the back of the mount and the bottom of the round mortise, or recess, in the chair back. I use a small firm cushion to elevate one end of the back to near level. Put the mount into place and twist it to a fro a bit to seat. Line up the impression of the bracket and place a weight, like a soup can, on top. You don't need anything heavier. Things may want to slide around some, and it can be a balancing act, but be patient and you'll get it. Clean up any squeeze out before it hardens by gently wiping with the cotton swabs and alcohol this time. Remove the weight in the morning, but give it a couple of days at room temp. before use.
Bon Adhesion!

Thanks tktoo!
But I am a bit worried about removing the leftover rubber skin from the wood. I am afraid because it will be missing that piece it won't sit flush?
Methinks I should leave it and just clean/roughen it. Thoughts?
Or are you suggesting that I get a new mount?

Hmmm... I forgot about that.
I haven't faced that situation with my set. I am curious how you ended up with that much black rubber left behind, though. The shock mounts on my chairs are metal on the backs except for a small hole in the center and a very thin band of rubber around the edge. They all popped off clean! My repairs basically join wood to steel. A design mod perhaps? Can you post a pic of the back of your detached mount? I think it must be different.
I guess my main concern would be the bond between the PC-7 and the old glue. Without knowing what HM used, it is unpredictable. I'd be tempted to just wipe everything down with acetone as is and go for it. What's the worst that can happen? Remember, you're not looking to fill voids here, so be sparing with the glue. Don't starve the joint, make sure you've got complete coverage, but it's sure easy to make a mess.

Drill, baby, drill...
No worries...

rubber
The glue clearly will bond rubber to rubber as it does rubber to wood. It seems you would be fine to use the old. But if you feel it a good idea to clean up the old rubber from the wood, you can buy a replacement shockmount from HM or on ebay from Special K. They make a good product.

I'll post a photo of the rema...
I'll post a photo of the remaining mount's backing tonight. I think it still looks like it's rubber covered. The rubber left behind is pretty minimal. A pretty thin layer and not all over at that.
Thinking I want to use the original mount if possible. COuld get a new one yes, but I am not sure that it is necessary. If it pops off again I am sure I will change my opinion on that.