Ten Thousand Cranes
I know this is very belated, but I hope I'm not too late to add a few pieces of information on this subject
As has already been said, cranes in Japan (and also, I believe in some other countries) are a symbol of long life, and consequently of prosperity and good fortune. But the real, living crane is actually a long-lived bird.. There have been records of the Japanese Crane living for eighty or more years in captivity, and while it is unlikely that this would be equalled in the wild, they still live for a very long time. Incidentally, what an exceptionally beautiful bird the Japanese Crane is. Fortunately it is now protected and its numbers, according to the last report I read are increasing significantly.
The main rival of the cane as a symbol of long life in Japan is the turtle or tortoise. We all know that the giant tortoises on various isolated islands in the oceans live extraordinarliy long lifetimes -well over one hundred years and perhaps two hundred: records don't always go back far enough to be sure. Whereas in Japan the crane is supposed to live for one thousand years, the tortoise is attributed with ten thousand years. The crane and the tortoise are often represented together against the background of a Horaizan, a mountain representing a utopian land of perrennial youth and immortality.
Isao Honda gives a paper-folded version in his book on Noshi. His version is in the form of a wrapper to contain gifts such as of money at times of celebration, such as at weddings.
Why a thousand - why ten thousand? Obviously, in the first place, because they are large numbers. But why multiples of ten? Again, because we have ten figures and since primaeval times have learnt to count in tens .Hence the decimal system.
But ten is an awkwrd number: it divides only by two and five, which isn't particularly helpful. Perhaps it would have been better if we had been designed to have six fingures on each hand. Then we would be counting in twelves. Twelve can be divided by two, three, four and six, which would be much more useful. Let's all join the duodecimalists!
Of course, in practice, conversion to counting in twelves would be impossible: the decimal system is much too entenched. The change would be even more difficult than fixing the date of Easter or changing the QWERTYUOP typewriter keyboard which was not designed for speed of use, but to prevent the early typewriter levers from jamming together. Even now, although we use computers that don't have levers, we're still stuck, but now with an inefficient system.
Cranes have only four "fingers" on each foot, so presumably they count in eights!