Random Choice interpretation Due to McFadden Economists usually interpret Random Choice, in the population sense, as each DM being drawn from the probabilistic choice rule independently and identically. However, Psychologist have maintained that individual DM are actually stochastic, in the sense that when presented with the same menu several times they would choose different alternatives. I am not fully convinced by either way of understanding probability in Decision Making, is there any work on understanding this, neuroeconomics, epistemological or anything related. 
In short, what is the interpretation of random choice for an individual DM?
I don't feel these two interpretations are mutually exclusive, they belong to different sides of the same problem - one is empirical the other is theoretical. The conflict you seem to see between these two methods is that the "population" interpretation is individual specific. But the population component is a statistical convenience. It is "everything else" that the researcher has no time to model, and it is variation on the level of the population of observations not people - so it is not an individual specific error. 

In fact, one acceptable interpretation of this convenience is that conditional on non-random components individuals are the same, and the errors are such that if presented with the same decision again the individual would make a different choice. This is precisely equivalent to your neurological interpretation. Extra structure on the random component may change the degree to which this justification applies - if errors for an individual are assumed to be correlated over time it decreases the relative impact of a neuroeconomic explanation and starts to be about unobservable heterogeneity on the level of the individual. But random utility can still easily encompass both explanations as answers to different sets of questions. The residual error will include both neuroeconomic reasoning and any variables that have been omitted.

To the extent that the randomness is specifically designed to capture "everything else" talking about what causes it is something most empiricists don't want to focus on. This being said, Michael Woodford has written something recently about neuroeconomics and random choice. And the "stocastic neural functions imply stochastic choice" is apparently a fairly common assertion in neuroeconomics: (see here, here, and here though some of these references point out that neuroeconomics places constraints on the form of random choice). But given the variables omitted in an average econometric study, I would hazard a guess that the random term captures relatively more of this omission and less of neurological processes.