It's a little unclear what you're asking. In general, psychologists try to build models that are parsimonious; this often means only introducing new parameters (particularly free parameters) into a model when they are absolutely necessary. You are right that with a sufficient number of free parameters, one can build a model that fits the data perfectly. But such a model has no generalizability: if you overfit your model to your data, the model will predict novel data very poorly.
Additionally, a model is preferred when its parameters have some psychological plausibility. That is, a parameter of the model should correspond to some psychological construct (e.g. working memory capacity, neural noise, etc). This is particularly important because it tells psychologists something about cognition and behavior, which is often the goal. If the parameters of a model are completely arbitrary, it's not always clear what we learn from it.
You are again correct that the choice in how to model psychological data is still vague. One many choose between any number of formalisms, such as neural networks or symbolic systems, and it is common for a phenomenon to be modeled in many different ways. Indeed, much of the cognitive modeling literature is full of numerous approaches to the same problem, with each author arguing in favor of their own approach.
The best way to figure out how to model your data is to read other papers that model the same or similar tasks. Look at what other researchers have done, and assess the merits of their approach. If possible, read multiple papers that model the same task in different ways (often researchers from opposing camps will cite each other).