Just to clarify off the bat I don't mean watching cut scenes with your friends, but rather cut scenes that can include them.
Something that's always bothered me is cut scenes where you're with your party doing something (an alliance raid) and then the instance ends and suddenly it's just you and the main story characters. What happened to those other 23 guys? Or even just the other 3?
So what I'd like to suggest is cut scenes where you can have your friends/party members in them. Not that they would change the cut scene in any way other than being there. Sorta like the cut scenes you get during instances where it shows the party members all readying for battle or after victory, except during regular content. Doesn't have to be old content, possibly just new stuff too. Also wouldn't have to include everything either.
And to clarify even further what I am mostly suggesting is to add 3 or more 'cut scene slots' where you can change or modify a couple of characters in your cut scenes. If you leave them empty, they just get filled by generic flames/maelstrom/adder characters or you can specify that you want them all to be a specific GC or maybe even to have your GC garrison guys fill the slots and you can specify which ones (or random if you don't care). Possibly even your chocobo or even allow players to pick up cut scene appearance characters.
Unfortunately I don't know how the cut scenes are made, but I assume they must have some level of dynamics since your current outfit is always shown. I assume it'd be possible to make placeholder spots that are just filled by 3D models that are imported into them. In this way I think it would be cool to be able to add others to your cut scenes and save their appearance (or just pull whatever they're currently wearing) and simply pull that appearance for that particular slot.
And if you don't want to be in someone else's cut scene it could just be set up so you need to ask permission by sending something like an 'invite to join cut scene party'.
In terms of actual usage, this could range from them just being background characters who hang out and don't really do anything, to being actively involved in the cut scene in some way. For example, you could have 'role based' slots where you can put a friend's tank class in the tank spot, then another friend who is a dps in the dps slot and a healer friend in the healer spot. Then during a cut scene, those roles could act as they would in their role. It's not like they have to say anything either.
It might even be nice to be able to swap yourself out of being the main character and let someone else do it. I personally would prefer to make my twin the main character as I don't really find myself acting much like they would to be completely honest. Really this is just an idea I simply had to get off my chest and I have not seen anyone suggesting it (though I am fairly poor at being able to browse).