Re-wiring a Nelson bubble  

Who's "they"?
If it's currently working, and there are no exposed wires, then why rewire?

Why rewire.
There are a few reasons:
1. No VISIBLE exposed wires doesn't necessarily mean no exposed wires. Old insulation can crack or fray, solder joints can crack, wires can short against metal parts... And it's possible for all that to be invisible. Fortunately, you can test for it pretty easily: Just plug the lamp in, turn it on, and touch all the exposed metal; if you die, there was a hidden short.
2. The old plug probably isn't polarized -- i.e., its two blades (I'm assuming that this is a US plug) are the same size -- so it can be inserted in the wall socket either way. If the plug is turned the wrong way -- and there's a 50/50 chance it will be -- there are two potential issues:
a) If there's an inline switch on the cord, it should be on the "hot" wire. If, because the plug is inserted backward, it's on the "neutral" wire instead, then even when the lamp is "off" there will be potentially deadly voltage at the socket. Fortunately, you can test for this pretty easily: Plug the lamp in, turn the switch off, and put your finger in the socket; if you die, the plug was backward.
b) Even if there's no inline switch, it's desirable for the hot wire to lead to the protected "button" contact at the bottom of the bulb socket, and for the neutral wire to lead to the more-exposed threaded shell of the bulb socket. If this arrangement is reversed because the plug is inserted backward, then changing a burned-out bulb while the power is on can be dangerous. Fortunately, you can test for this pretty easily: Plug the lamp in and accidentally brush against the edge of the socket while changing the bulb; if you die, the plug was backward.
3. Even if the plug is polarized, you don't know who last rewired the lamp or whether they did it right. If they did it wrong, then you'll have the above issues EVERY time you plug it in, not just 50% of the time. Again, you can test for this... Etc.
The safest thing to do -- although few people do it with portable lamps like yours -- would be to rewire the lamp with a three-wire plug (hot to the socket's button, neutral to the socket's threads, and ground to the outer exposed metal bits) and a new socket.
Second-safest would be to rewire it with a modern two-wire polarized plug (again, hot to the button, neutral to the threads) on new wire with a new socket. The new wire and socket will protect you from most of the electrocution danger, and if your house wiring is correct, the polarized plug will protect you from much of the rest.
You can do all of this yourself -- it's not rocket science, just maybe a little time-consuming -- or you can get it done by a qualified professional for probably around $25, depending on how much cord you want for your swag setup.

Some lamps can be quite...
Some lamps can be quite awkward to rewire and seem to need 3 hands (Clay Michie for Knoll please step forward) but this should be very quick straightforward rewire

That's an easy re-wire. In...
That's an easy re-wire. In fact, Ikea sells a paper handing bubble lamp for something like $9. You can toss the paper lamp and use the included 6-foot white cord and socket. It's already wired. Then buy this cord restraint:
http://www.moonshineshades.com/index.php?page=shop.product_details&flypa...
You'll have to take the new socket apart and put it back together in order to fit it into the nelson bubble, but that's easy, just put the wires back where you found them--mark them with a sharpie if you like, to make it easier.
http://www.moonshineshades.com/index.php?page=shop.product_details&flypa...

I'm surprised more people...
I'm surprised more people don't have trouble with rewiring these. I had a heck of a time finding thicker pendant lamp cord so I broke down and bought the aforementioned Ikea pendant sockets. I'm having trouble removing the original socket. The bottom half doesn't want to unscrew or pull off and I'm scared of bending the bracket. It just spins along with the cap on top of the bracket. Anyone have experience with how these are connected?

I really don't
see the point in rewiring just to satisfy the rumblings of the fear-mongers always present in society. If it's working now, and no damage is evident, I think it can be safely assumed that it's not suddenly going to become a life-threatening instrument.
If a short would somehow appear, a fuse or breaker would trip, and that would be the end of it. If you really want to fantasize about electrocuting yourself, there are more effective ways of carrying out such a plan.
There are no moving parts in a plug-wire-socket-lamp assembly that has no switch. The fixture has presumably been hanging undisturbed somewhere for a few decades, and you plan to treat it similarly. With the kinds of repair service currently offered, and the increasing complexity of today's wiring devices, it wouldn't surprise me to find that you'd be worse off for having intervened. Sometimes "If it ain't broke. . ." really makes sense -- to me.

I have 3 bubble lamps and...
I have 3 bubble lamps and none of them have wiring at all -- just the socket. So, I'd say rewiring is kind of necessary.

Are you kidding, SDR?
The lamp is most certainly a ticking bomb!
Really. Just LOOK at it, man. That thing is liable to self-combust at any moment!
I wouldn't even have it in my house.

Well -- it
DOES look a bit like a harbor mine, now that you mention it.
Maybe we should go back to whale oil and bayberry candles ?
Parafo, sometimes there are screws inside the socket, next to the bottom contact, that will allow the socket to come apart. At lease I seem to remember such a thing.