Any Ideas How to Make This?  

Maybe try some plasticoat or...
Maybe try some plasticoat or heavy shellac and string or something?
If I were trying this, I'd use a giant beach ball instead of a balloon. If you do use a balloon, try and make some different shaped versions.

I've made dozens of these--
My mother used to make small versions as Christmas decorations, I've made large 2'-- 3' versions for use in window displays.
As a stiffening agent, I always used wheatpaste, brushed onto the string wrapped balloon. If you want something a little more brittle & lasting, perhaps you could then spray the resulting string ball with spray polyurethane. (AFTER popping)
BIG TIP, that'll save you grief: when the string has dried, BEFORE popping the balloon, be sure to separate the string from balloon surface-- they're kinda stuck to each other, at this point. Poke (with fingers, not sharp object!) the balloon's skin away from the string web, through any openings. You'll hear the bond separate as you do it.
If you fail to do the above, the string web will violently shrivel up like a deflated balloon.

Just thought of something:
Since this lamp design requires a large circular hole (presumably, to access the socket & bulb), maybe you should draw an appropriate-sized circle onto the balloon before wrapping, to use as a guide. Tracing around a bowl in felt-tip would be an easy way to do this.

Just thought of something else:
(sorry to bloviate-- I'm a rabid crafter!)
Maybe you should also draw a nickel-sized circle to approximate where the cord will enter, then wrap the string to build a similar hole-opening. It should make for a stronger support for where the globe's weight will hang.

punchballs
Punchballs are round when inflated (unlike balloons, which are more pear-shaped). They would make a good mold.
Epoxy is really messy to work with. The commercially made ones are probably made by machine, or at least with the assistance of machines. Epoxy is gooey and sticky and tends to get everywhere you don't want it to be.

Extra large balloons
like the type one would buy from a party-supply store, are round. But, I see no reason punch balls wouldn't work.
Good luck-

My two cents...
Drawing a circle on the ball won't really help you hold a circle open. I'd glue a collar onto the ball using carboard or something so that you can wrap aroundit as you go. You can then use that spot hold the blow-up nozzle so you can deflate the ball.
As for the stiffening agent, I'd be afraid that wheat paste would A)dry out and crumble over time and B) allow the ball to sag as the paste never really 'cures' and will absorb humidity and then allow flexing. I'd use a good quality cotton twine soaked in a bucket of slightly thinned wood glue. The wood glue will tint the string a bit on the yellow side, but that'll actually look quite good. If you want a clean white, I'd use plain old Elmer's. If you wanted a more rustic wabi-sabi look try using a jute or hemp cord, both of which are a nice brown tone.

Mod-podge
Mod-podge, a decoupage glue, might work better than Elmer's or yellow wood glue. It's thicker and might dry harder, I dunno. There's also Aleene's Tacky glue which is like Elmer's only a lot thicker. There are a bunch of other white glues, too.
I agree on the wheat paste. It wouldn't work here in humid Baltimore with no A/C, plus bugs would enjoy feasting on it.

PVA
glue ("white glue") dries as rigid as yellow glue, and has a longer open time. . .
I wonder if Silly String would work ? Nah. . .
If a slightly tacky string were wound "dry", then one or more applications of thinned white glue could be sprayed on the ball afterward, to weld the thing into one piece.
I say listen to Holden -- he's actually MADE these. . .!

When i first saw that lamp...
When i first saw that lamp i wondered how i would make it. Most ideas posted would
work. The holiday crafters have been proving it for years with the stiffened crocheted
holiday things. Most white glues are water soluble, not all. Most commercial
wheat pastes and clear vinyl pastes have insecticides. That is why they are considered a bit toxic and most
containers suggest gloves. Also water soluble, but a top coat would lock in the shape
and seal it. As Williams success states.
The epoxy fiber would have a nice luminous quality, but I would experiment with the twine
and glue method first. Super cheap materials and would be good practice in getting
the wrap quality of the string looking even. Someone in my neighborhood had a
dozen hanging from a tree over the holidays. Looked great.
The epoxy method is doable and may find what you need at your local 'hobby lobby'.
Wonder if a two/part clear cast would work. Mixed in a deli-type container with a lid
and small hole in the top to draw the silk-type fiber through. (boat building suppliers
have the correct materials). Anything two/part is crazy toxic. If i went that direction i
would do it outside and let it cure outside or a very ventilated area like a garage.
'Pour on' and 'clear cast' from the hobby store fit the toxic category. Their glue section
is an isle long and the thin silky thread for doily knit craft is another isle long. I'd read
the labels and get the best glue that is not flexible when dry, dries clear, and water tight.
Epoxy and fiberglass thread does not seem necessary. Not worth the health risks.

Thinner string
I'd use thinner string than yarn, something like a thin cotton butchers string. And I wouldnt use wheat paste as it can re-activate if it gets wet. Even humid days can affect it.
Also - make sure you use some kind of release agent on the ball/balloon. Otherwise you might spend a month snipping little bits of balloon off of it with scissors. Something like vaseline will work.
I also agree about avoiding epoxy - nasty stuff that will fuck your brains over if you dont use proper ventilation. We were forbidden from using it in art school for precisely that reason. However, the originals are pretty sturdy and flexible - if they get compressed the form will pop open again.

Fiberglass yarn
Wax the balloon/ball, get some fiberglass yarn, and dip the yarn is resin. Then add a mess, stress, fun, the word tenacious, maybe a helper, and your credit card(just in case your results warrant buying a real one). I've wanted to make one for a long time.
http://www.fiberglassking.com/fy.htm

Don't listen to any of them, Sammyman--
They're leading you down a sticky path to disaster, which will sour your taste for crafts-projects forever after.
Wheatpaste. Spray polyurethane.

Poets
we finally got poets. It's about time. . .
I've assumed this was to be an indoor object.