Using hyperpriors only makes sense in a hierarchical Bayesian model. In that case you would be looking at multiple groups and estimate a group specific coefficient w_group based on group specific priors, with coefficients drawn from a global hyperprior.

If your prior and hyperprior reside on the same hierarchical level, which seems to be the case you are think about, then the effect on the results is the same as using a simple prior with a wider standard deviation. Since it still requires additional computational costs, such stacking should be avoided.

There is a lot of statistical literature on how to pick non-informative priors, often theoretically best solutions are improper distributions (their total integral is infinite) and there is a large risk of getting improper posterior solutions without well defined means or even medians. So for practical purposes picking wide normal distributions usually works best.