Seconding Tevish. Give a dwarf a barrel, he drinks for a day. Teach him how to brew...
Designing a panic room is really very similar to your first steps of settling a new area. The priorities are the same. So deciding how to stock it is easy-- just stock it with whatever you need for new embarks.
A walled off section of cavern is the best place for a saferoom, because it has good soil, and likely the potential for some silk. Make it as big as possible (for trees, mostly), then leave a pick, an anvil, and an axe in there. Even if it's deep, it's not really far from the surface-- just ten or twenty steps up a staircase.
Don't build or dig too close to it. The design is such that if your panic-room dwarves need more stone, they dig for it. If they need more food or booze, they grow it. If they need more wood, they chop it down. If there are spare dwarves and they need more cavern, they build an airlock and try to claim some of it. The idea isn't that the panic-room dwarves have everything they need-- it's that they have the capability to build everything they need. Just like starting a new fortress.
If you want to go from there, a safe water source is a priority, and not too difficult to set up, not in most caverns. A little bit of stockpiled (and probably forbidden) food is a good idea. As many slabs as possible are a good idea. Moody dwarf traps aren't a bad idea, but they're not necessary. If there's more than one dwarf in your panic room, though, all you have to do is get that dwarf to build a workshop that can be walled off, and if there's only one, you don't really have to worry about it anyways.
All your panic room dwarves have to do is eat, drink, and not go crazy. Anything else-- danger room training, armor, workshops-- they can build as they see fit. If you want greater survivability, you can get it by having a larger number of panic rooms, with independent populations.
Usually, if a fortress is close to falling, the outside is no longer safe, but in certain circumstances (say, drawbridged single entrance, and you fall to a misjudged deep dig), migrants can be very survivable, since they don't really know or care about any of the dying inhabitants. It's not a bad idea to make a panic room accessible from the map edge. There are Roach Hotel designs (dwarves get in, but they don't get out) that can make these useful even if the outside is very dangerous. Civs trigger pressure plates can be very handy.
The best thing to do, of course, is to think ahead, and have those panic rooms staffed by dwarves who have never met any of the other fortress inhabitants. That gives you a good head-start on the happiness problem. The only major concern besides happiness, food, and drink is ghosts.