Over 150 years ago in the "free banking era", any bank could issue its own banknotes. You would see a number of different notes in circulation - you could have $2 from The Bank of Chattanooga, The County of Polk, or The Lawrenceburg Bank of Tennesee. Same in Canada. While it created a lot of hassle for anyone wanting to use the currency, especially if they would travel beyond where those notes would be redeemable, it also allowed for a market to form and put a real value on the currency based on how credible the issuing bank was - good notes would be valued at par, bad ones - at a discount.
While banking nowadays is certainly simpler and safer with uniform currency issued by one entity per country, FDIC deposit insurance to prevent people losing money in case a bank goes bust, etc. However, this means we also lost the ability to evaluate bank's trustworthiness, usability, etc. and arbitrage it.
In the Bitcoin world, it is fairly straightforward to judge the health of an exchange by looking at its exchange rate. Back when MtGox was going bust, its rates deviated from its competitors by 15+% even early on. When withdrawals out of the exchange became impossible, there was even a secondary market that traded MtGox BTC for real BTC by using MtGox's inter-account transfer capabilities. You can track arbitrage metrics today.
How would this apply to banks? Well, you could start with the currency issued by the government as the base - one dollar here would be redeemable to one dollar in banknotes (this is what MintChip aimed to do for example). Banks would use that as their reserves for fractional-reserve banking and issue their own debt-based currency. All of it could be tracked on a shared "bankchain" to allow market for various bank debt to form. The price difference of the debt could stem from various factors - how stable the bank is (FDIC insurance is all well and good, but nobody wants to go through the stress of having your saving locked up for who knows how long), how cheap and easy it is to transact to and from a bank (say, USD-demoniated bank account in Europe might be valued less due to the extra cost of transferring money overseas), how accessible is the bank (branch opening hours and how common they are), as well as how their customers share the banks values (banking for millennials, sharia-compliant finance, etc.).
Setting up such simple metric for each bank would allow anyone to easily compare various banks and put that metric on everyone's mind. If one day your money would go down to 95 cents on the dollar, perhaps you would ask your bank "what is going on?" and find out that HSBC enabled Mexican drug cartels to launder money. Maybe it would enable some people to demand all of their deposits to be covered 100% by the government-issued currency, rather than allow for fractional reserve banking? Or perhaps it would allow some people to move their money to their local credit union to support the grassroot company and earn 1-2% on the conversion rate.
Banking in the modern world is pretty homogeneous - currency dictated by the government, fungible money no matter where you go. Perhaps it might be useful to bring the market back into the equation and allow us to see see the bank's worth by checking the value of their dollars?