Bootstrapping as a way to deal with multilevel data (similar to Bayesian hierarchical models) Suppose you want to estimate the linear relationship between $x$ & $y$ among people. You can't afford to ask N = 100 people to measure the variables of interest and estimate the relationship, so instead you ask N = 20 people and you take repeated measurements from each individual (100 observation per person which makes a total of 2000 data points). But now your observations are not independent and are grouped with respect to individuals. You can run a linear regression for each individual separately and the relationship can be expressed as:
$$y = intercept_{j} + \beta_{j} \times x$$
$$j = 1,...,20$$
So you will have 20 different relationships ($\beta s$) which are individual specific but you are interested in the $\beta_{population}$ or the relationship in the population. Having a set of $\beta$ s ($[\beta_{1}, ...,\beta_{20}]$) I would take bootstrap samples from them (let's say 100 bootstrapped sample of the same size) and calculate their means to estimate the distribution of the $\beta_{population}$ (assuming $\beta_{j} \sim N(\beta_{population}, \sigma_{pop}^2)$) with its confidence intervals. Would this approach be reasonable or its not valid in ways that I can't see?

Another approach could be to draw one X×Y pair from each participant, construct β from those 20 points, and then repeat that a large number of times to get a distribution of β.
What concerns me about the method you proposed is that it seems to me you're throwing out information about the variability of the $\beta_j$.
One fairly straightforward way to test the validity of either approach is to make an artificial population, get the sampling distribution of β for that, and then try whatever method you have in mind drawing 20 from the full population and see what happens.
