History of unstable formulas There are many equivalent definitions for stability, one of them being that being unstable is equivalent to the existence of a formula having the order property.
While intuitively it makes sense that having a lot of types (which is the definition that Shelah uses) would lead to a lot of models , the order property isn't as intuitive. Where the examples/partial results floating around that would have indicated that it was likely? The only result I'm aware of is Ehrenfeucht result, which said there were at least two models (for uncountable cardinals).
Edit1: I'm unsure why this question was tagged as unclear. If someone could explain why so, I can edit it. 
 A: I disagree with the claim that the order property isn't intuitive.
When I took my first model theory class, types were introduced to me as a generalization of Dedekind cuts. So the very first examples we looked at were linear orders. Moreover, the construction of theories with exactly $n$ distinct countable models (for $n\in\mathbb{N}-\{2\}$) was done via linear orders with additional structure; I believe this is standard. So we were already used to counting the models of theories with linear orders. 
Given this, it was very intuitive to me to believe that having an ordering meant having lots of types. Now, that's not actually true - DLO has the order property obviously, but only $n!$-many $n$-types - but it's a first approximation of a true fact: that having an ordering means having lots of models. So for me this was the progression of ideas.
Although I describe above the way I gained an intuition for the order property, Shelah et al were certainly aware of all of the above and more. Finally, there were already at the beginning of stability theory lots of set-theoretic results about the different kinds of linear orders, so there was already a strong sense that $\{$linear orders$\}$ is an extremely rich class. So I think the order property was actually a very intuitive thing to define in the context of counting uncountable models. The fact that it corresponds exactly to instability is less intuitive of course, but the order property as a thing to study in the context of stability theory seems natural to me.
