There are many solutions to the problem that typically not enough information is available to fully specify a prior.
For all approaches (I know) but weakly informative priors I kind of understand why they are reasonable solutions to the problem: "There is not enough information to specify a precise prior." For weakly informative priors this connection escapes me. To give an example, let's say I want to estimate the mean of a Normal distribution but have absolutely no prior information. Using a typical weakly informative prior $\mathcal{N}(0,\sigma^2)$ with $\sigma$ being large, I do not see how this is an implementation of no prior knowledge. Instead, it roughly corresponds to any value being equally likely, which is completely different compared to complete uncertainty about the prior as is for example achieved by minimax methods.
Even in the other situation where weakly informative priors $\mathcal{N}(0,\sigma^2)$ seem to be often used, namely for implementing the prior knowledge that deviation from $0$ become equally unlikely weakly informative priors seem to be to optimistic in the sense that they assume a fixed $\sigma$. Alternatives like estimation $\sigma$ (Empirical Bayes), putting a hyper-prior on $\sigma$ (Hierachical Bayes), considering all $\sigma$s a possibility without assigning each $\sigma$ a probability (Gamma-Minimax) seem all to be a better suited.
At the same time, weakly informative priors are recommended by many experts (for example, in https://www.amazon.de/Bayesian-Analysis-Chapman-Statistical-Science/dp/1439840954) as a good solution to the problem that there is not enough information to specify a precise prior.
So, what am I missing? I have looked in https://www.amazon.de/Bayesian-Analysis-Chapman-Statistical-Science/dp/1439840954 but the arguments presented are more in terms of favorable properties (regularizes, weaker than the actual prior, let's the likelihood speak) and not why this approach is a good approach for implementing uncertainty in the prior and should typically preferred over alternative approaches.