How Old Is Jujitsu
In fact, there is no single source of Jujitsu. Its history is one of convergence not divergence. The best answer to the question "How old is Jujitsu?" is simply "It depends on what you call Jujitsu". There is no magic syllabus which contains it and nor has there been a time when people stopped adding or discarding techniques. Another problem, is that some people use the term to describe very specific schools while other people use it as a general term for all the Japanese fighting arts.
What we can say is that humans have always fought each other. So unarmed combat has always existed in Japan as it has everywhere else. It first touches history in "The Chronicle of Japan". This was written in 720 AD and mentions a contest of unarmed fighting taking place in 230 BC. This event is claimed as the beginning of Sumo. But this is not really evidence of the systematic study of a recognisable system. For evidence of that we have to wait for the Nara period era. This lasted from 650 A.D. to 793 A.D. and marks the emergence of the armed gentry (bushi ).

The Beginning of the Samurai
Members of the bushi went by a great many titles and held many social ranks, but the word "samurai" has become so widely used as a blanket description for all of them that I will continue to use it here. The word comes from saburaui which means service. So from their very beginnings, the samurai were defined by what they regarded as their virtue - loyalty to their masters.
At this time, they were mainly land owning farmers. Battles were rare, but when called upon by their masters they fought mainly on foot with spears and were rewarded with land.
By the time of the Hain period, things had changed. Battles were much more frequent and the samurai had emerged as a distinct class. Training for war was now a full time activity and many different schools of study (ryu ) existed.
The following centuries were characterised by more and more civil wars. The samurai became more and more powerful and rose in social status accordingly.

Early Techniques
It would be wrong however to believe that any these ryu corresponded to any modern martial arts. The main difference is that the division into armed and unarmed techniques is a modern one. The study of battlefield weapons was of paramount importance and "empty hand" stuff was mixed in with these.
For example, atemi was practiced with the butt of weapons and much as with fists. Wrist locks were studied by swords men to deal with grabs to wrists and handles.
It is from this period though that many that many characteristics of jujitsu emerge.

Throwing
In battle the samurai wore armour. This means that punches will cause no damage and kicks will be slow and clumsy. So you throw. Armour offers little protection against blunt impact damage and makes you a very easy target on the ground.

Locking
Armour also fails to protect the joints. So breaking joints is another good way of causing damage. Crude arm and head locks and strikes into joints would have been important.

Disarming.
If you are unarmed, the chances are that everybody else isn't so obtaining a weapon is vital. This is where wristlocks come into their own.

Atemi.
As already mentioned, weapon training involved all of the attacks a weapon can make, not just those it was designed for. With everybody so well protected and mobile, accuracy of strikes is far more important then their power.