What is the stochastic part in stochastic gradient descent? From my understanding, a stochastic process whose value at a particular instant is dependent on the previous values taken and every time the process is run, the path chosen can be different. We can know only the limits and confines the process stays after certain intial seeding value.   

The weight values for a neural networks are updated via the stochastic gradient descent method. What is the stochastic part about it? After some initialization of the variable, error function accumulated will be same if the same set of input data is provided after initialization in each test. Why is the relevance of the term stochastic? Where is the scope for the randomness and what is the source of this randomness?
You sample a random mini-batch as opposed to the full dataset. This means that you get a stochastic approximation of the true gradient of your loss function with regards to your dataset and weights.