What is the origin and purpose of the multiple hand symbols that Ninjas [Ok, Naruto and the Protagonist of Ninja Assassin] make?
Likely made up shit. Pretty much all “Ninja” lore is fairly recently made up, backed by a few facts. Ninjas were a *Secret *Society.
wiki "*The origin of the ninja is obscure and difficult to determine, but can be surmised to be around the 14th century.[4] Few written records exist to detail the activities of the ninja. The word shinobi did not exist to describe a ninja-like agent until the 15th century, and it is unlikely that spies and mercenaries prior to this time were seen as a specialized group. In the unrest of the Sengoku period (15th - 17th centuries), mercenaries and spies for hire arose out of the Iga and Kōga regions of Japan, and it is from these clans that much of later knowledge regarding the ninja is inferred. Following the unification of Japan under the Tokugawa shogunate, the ninja descended again into obscurity.[5] …Ninjas figure prominently in folklore and legend, and as a result it is often difficult to separate historical fact from myth. …Despite many popular folktales, historical accounts of the ninja are scarce. … Historian Kiyoshi Watatani states that the ninja were trained to be particularly secretive about their actions and existence:
"So-called ninjutsu techniques, in short are the skills of shinobi-no-jutsu and shinobijutsu, which have the aims of ensuring that one's opponent does not know of one's existence, and for which there was special training."...."*
They weren’t even primarily assassins "Espionage was the chief role of the ninja"
As to “hand symbols” are you talking about a sign language type thing (which no doubt they used, but we certainly have no record of) or hand “seals” ?
wiki "*Kuji-kiri is an esoteric practice which, when performed with an array of hand “seals” (kuji-in), was meant to allow the ninja to enact superhuman feats.
The kuji (“nine characters”) is a concept originating from Taoism, where it was a string of nine words used in charms and incantations.[95] In China, this tradition mixed with Buddhist beliefs, assigning each of the nine words to a Buddhist deity. The kuji may have arrived in Japan via Buddhism,[96] where it flourished within Shugendō.[97] … Later, the use of kuji passed onto certain bujutsu (martial arts) and ninjutsu schools, where it was said to have many purposes."
*
They probably originate from Hindu/Buddhist mudras via the time-honoured Japanese tradition known as Stealing Shit From The Chinese & Making It Look Cooler
According to one author, whose name escapes me at the moment, when you are under stress, you go through the gestures in a specific sequence. The ritual helps you remain calm and concentrate on the task at hand.
And, if your enemy sees you calmly performing the ritual when you ought to be panicking, it might make him nervous.
The origin of the Ninja is modern film era.
Anything prior to that is manufactured history - or history interpreted in such a way as to give ninja’s history. (Some of the absorbic self delusion includes claiming that it always existed (since the 14 th century but the name changed. They have a book of teachings but only one man is ever allowed to see it…etc…)
Therefore Ninja Hand signals originated in the movies in th 1970s
But they didn’t make them up out of whole cloth. As we’ve shown, they *originate *in Japanese Buddhist practice and folk magic, not the movies.
Thanks, I now know how to start (or end) a conversation at the next martial-arts film I attend 
This.
Thanks.
The worst part is that they weren’t even all that secret, and many of their “ninja techniques” were just plain common sense or tricks that took advantage of the weird holes in Japanese styles of warfare (sometimes literally: Japanese seige techniques were ludicrously vulnerable to infiltration or sabotage). Originally, they were semi-secret because of their isolation, but the techniques spread enough that numerous daimyo took advantage of ninja.
Of course, they also weren’t a society. Ninja were warriors trained in many different villages, and when they alter on more or less joined together to support Tokugawa Ieyasu, it was largely a recognition that Ieyasu valued them. Thus, many of the ninja did operate under his banner; Ieyasu is also the only japanese leaer to ever formally commend his ninja for excellent service. But afterwards they had no need to remain banded together and tended to sell their service as spies and counter-spies all over Japan.
True, much about ninja is popular culture. But in my opinion, whatever ninja truly were adding Chuck Norris-level urban legends surrounding them certainly adds to their job.
Let’s say, for example that ninja were simply a particular type of guerilla fighter. If your adversaries think you can disappear in a poof of smoke, jump around like a kangaroo dressed in all black, throw pointy whirly pieces of metal at people, make explosions/tornadoes of leaves, and have all sorts of crazy killy stealth gadgets and ninja juju it just makes your task easier. People will fear the mystique.
If I’m part of an organization whose goal is to destabilize a government, I want people to think I’m some mystical unstoppable force, because what I actually have to do to get the job done amounts to very little, and also any incidental accidents/occurrences (say, lightning strikes a bunch of flour sacks and causes a hellacious explosion) people will say, “Ninja did it” 
While there were legends about such things fairly early on, there’s no real evidence that people thought the ninja were really magicians. Or at least, the employers certainly didn’t.