The one feature I want most from a bank would probably require overhauling the entire system. It would be as follows:

I make a purchase, and within seconds I'm prompted on my smartphone to put it in one of my custom budget categories.

This would utterly solve the pain of sorting it out later, and have the added benefit of fraud protection. "Why, no, I DIDN'T buy gas in Ohio 10 seconds ago."

Super bonus points if the categorization data goes to an endpoint of my choosing, so I can ensure it's not used for data mining.