Time Series Forecasting, why try and predict randomness? Maybe I have a gap in my understanding, but I've looked everywhere and haven't found a satisfying answer. When we are doing time series forecasting, we need the data to be stationary, if not we apply various techniques to make it stationary. Then we apply AR or ARMA model. Ok now comes my issue, I understand that there may be memory in this stationary data, but isn't it stochastic? Why are we trying to model the randomness? I feel like it's useless, why try to predict something random?
 A: The idea is not that you're predicting pure randomness. You're predicting a process that has some kind of pattern.
Sure, there's the error term. We have error terms all over. We have error terms in regressions yet are perfectly comfortable making predictions from OLS equations. 
It's different if you wind up with a totally random process, and you might, but the hope is that you have some kind of pattern.
A: In practice, we are usually trying to predict some time-series quantity in reality and our statistical model is merely a tool to describe the behaviour of that quantity.  Thus, a priori, we do not know if there are patterns or aspects of the series that will allow viable prediction.  The ARIMA model is just one model we can fit to our data to see if it is a good description of the data.  If it fits well, then this suggests that there are some auto-correlation and moving-average parts in the data, and so it behaves differently to a "purely random" (i.e., IID) series.  The usefulness of the predictions will depend on their accuracy, and the value of being able to make better predictions about that quantity.

I understand that there may be memory in this stationary data, but isn't it stochastic?

Predictions of time-series retain the random element in the series, but it may take advantage of inferred auto-correlation or moving average parts.  Yes, the predicted series is still stochastic, but by modelling it with past data we may be able to make a prediction that is still better than guesswork for an IID series.  For example, if we have strong evidence of positive first-order auto-correlation in the first differences, then we would expect that high gains in the series value will tend to follow other high gains.  This is still stochastic, but we are now able to make better predictions than we could if we were dealing with an IID series.

Why are we trying to model the randomness?  I feel like it's useless, why try to predict something random?

Well, let me ask you this.  Would it be useful to be able to predict the value of Amazon stock tomorrow, or next month?  Or would that prediction be something useless?  Would it be useful to predict whether mean world temperature will be higher in fifty years than it is now, and how much?  Once you start asking applied questions like this, I think you will see that there are many manifestations of time-series in reality that would be useful to be able to predict.  It would be great if we could predict these things well, but even being able to predict them slightly better than pure guesswork is somewhat useful.
