Great! Finally a place for the crazy ideas!
Lending/Saving idea - Liquid banking.
Instead of the balance being stationary until a payment goes in or out (boring list of credits and debits as it has been since the Medici) - give users the option to drip a payment over time and have their money in constant motion. E.g Direct debit of £30 taken on 1st of the month - Merchant gets their money as requested but from Monzo instead of account. Monzo break the 3000pence into the seconds in that month and drip the repayment each second so balance is in constant motion. Monzo could charge interest on the average amount owed during the month. Likewise you could drip your salary in so instead of getting the lump sum at end of month Monzo could grab it and drip it in over the month - interest paid to user on the average credit during the month.