I've been thinking about that for a while. Besides a different design and time to reach feature parity, what would be the "unique" thing to solve the problem of managing one's finances?
I don't know if this exists (as someone who uses Mint but doesn't know the space well), but I would kill for a service that aggregates and simply reports my monthly bills for me. I don't need Mint's level of granularity and feature parity: I need a service that reminds me about the $5/mo I'm paying on a subscription that I haven't used in two years.
We do this rather well at MergePay (thanks for the mention twofishies) We give you a broad overview of you income, bills, and expenses daily/weekly/monthly and aggregate all your recurring bills and remind you either via email, SMS, or push if you use the mobile app, before they are due. I built it because I had the same issue.
As a hobby project, I'm working on a tool to help improve spending habits. I'm not crazy enough to imagine I'd ever reach feature parity with mint, but I personally know plenty of folks who use the well known personal finance tools yet seem to make a lot of careless purchases they later regret. And while mint is fine for what it does, I don't think it really points you in the direction of disciplined, thoughtful spending. If you want to stick to a budget but you find it hard, I think there's an opportunity for a more focused tool.
More than simply building a savings account, the other side of the coin is making your spending more meaningful by reducing impulsive buying. Saving a few bucks here and there with coupons or promotional discount is way too easy to spoil with just one thoughtless purchase.
I think gamification of "saving habits" would be very interesting - something to encourage you to save more. But it shouldn't take a lot of time to set-up or be confusing. If it would be automatic, even better.
The set up time is important- it seems to take too long to really get Mint set up to the point where it's useful, and then you still spend a lot of time recategorizing transactions so that you can know where your budgets really stand. I know that's not an easy problem to crack, but something with less detail that can encourage better overall spending/earning/savings trends would be pretty useful.