It seems that bounded rationality models focus on explaining a particular psychological bias, in a very specific way. In particular, its seems that the state of the art consensus is that one size does not fit all. The prevalence of framing effects makes this issue very difficult, but is there any way to think of a general approach to modelling bounded rationality. Is it regret minimization, or random choice, or rational inattention?
The term bounded rationality was introduced by Herbert Simon. He wrote
"The term, bounded rationality, is used to designate rational choice that takes into account the cognitive limitations of both knowledge and cognitive capacity. Bounded rationality is a central theme in behavioral economics. It is concerned with the ways in which the actual decision-making process influences decisions."
This passage seems to focus only on incomplete information, limited information processing time, limited knowledge -and not on any "psychological biases" like framing effects and the like. But the passage is complemented by a last phrase
"Theories of bounded rationality relax one or more assumptions of standard expected utility theory"
which suddenly opens the content of the term to anything. So "bounded rationality" has come to signify "deviations from strict rationality in any way", making it rather impossible to arrive at a general, all-encompassing, or even most-encompassing, modeling approach, that at the same time, will be specific enough so as to be operational.
One recent paper that is being positioned as a very wide-ranging theory of bounded rationality (although certainly it doesn't come close to capturing every insight in the field) is Gabaix's forthcoming QJE, A Sparsity-Based Model of Bounded Rationality.
Gabaix formulates a fairly general model where agents can rationally decide to pay limited attention to each economic variable (depending on its importance), and derives the consequences for the classic features of consumer theory, general equilibrium, and so on. Since this is not my field, I am not completely sure how this paper differs from the preexisting literature on rational inattention, but it does seem quite general and ambitious.
Herbert A. Simon coined the terms bounded rationality and satisficing. The connection between these concepts is that, since our cognitive aparatus has real-world bounds in terms of what can achieve (e.g. cognitive processing, memory, time, etc) people are often do not able to optimize. Hence they often satisfy with a sufficient good outcome (i.e. "satisfice").
The theory of expected utility still holds but people, instead of trying to maximize it, try to achieve at least a sufficient good level (threshold, aspiration level) in terms of expected utility. Hence, the first course of action that people find which is able to satisfy (be greater than) that utility level is then chosen to be applied.
Goal-programming is one of the existing decision-theory methods which is based on Simon's underlying principle.