Handling a GT is an age old topic and I think Brandons guidelines and all the info on this thread is great advise and we all come up with our individual processes on the boats to handle GTs. We have been lucky to be able to handle a lot of GTs and as with anything you get much better with experience.
But I also have to say that you can go overboard with trying to protect the GT and actually do more harm than good because you trying to put mats down, put hoses in the mouth plus trying to get cameras and getting people to help that are not experienced etc. I always find it quite amusing to see guys for the first time try and handle a big GT, they are usually docile on the boat but there is nothing like it no matter how strong you are. You have to be firm and make sure you handle the fish with enough strength to get the job done what you trying to do, e.g. if you are moving the fish, then move the fish don't half drag it and then stop and try go back to where the fish was etc.
Basically for us we try land the fish as quick as possible and find putting buckets of water through the gills straight away has helped hugely as there are no switches, trying to untangle hoses etc. Then making sure the deck or sling is wet and get the fish into position quickly for a picture, the trick is loading the fish onto the lap, this should be done with a towel on the lap (fresh not old and slimy as this is worse than bare skin) and then I put one hand into the fishes mouth and grip the jaw (or put my hand under the fishes head if small enough) and the other around the tail and do a dead lift onto the lap and the lifter must position the fish, the angler must sit and make sure with all his might that fish does not drop! It is all team work and don;t be scared to shout the orders to make sure everyone is working together. Carry on the whole time putting water through the gills.
The other options is an exercise mats, they work very well for captains foot sponge when skippering to reduce strain on the body and are great for protecting fish, but you need to be very careful with neoprene and exercise mats the material is actually quite course and non slip so will take slim off a fish, rather use PVC wet.
Hope this helps and my 2 cents.
Cheers,
Nick