Very impressive.
For now, at least, this sounds like a problem where human involvement is really useful.
Cranky Concierge, as someone mentioned below, is one great option.
Another is paying the ~$50 for lounge access for the day, to talk to the reservations agent at the desk inside the lounge. They're usually much more experienced, much more likely to spend the time to do creative routings, and haven't just had to deal with possibly several hundred angry passengers. In one case with United, I stepped out of line to speak to the gate agent after my flight was delayed several hours and paid to go in the lounge. I wound up getting the last confirmed seat on the next flight, whereas the person just in front of me in line had to go on the standby list (which she didn't clear). That $50 was _well_ worth it.
I've never had cause to hire them myself though.
I was pretty impressed with the service.
I forse a day when Google Now or Siri detects your flight delay and presents you with rebooking options via a push alert.
I'm a huge fan of tripit to parse my emails, and add all my travel details to my google calendar for me, but I happen to have never missed a flight and needed their service's help.
While perhaps impossible, I'd also love an app that would overall just help me get back ontrack for flights and rebooked somehow.
I realize I can go to the gate agent (who invariably now has 50 other people in line, and are working with computers that seem to require 1000 key presses to even look up the most basic thing) or call the airline (similar situation)- but it seems that the best results only happen for you if they aren't super busy and like you.
Oftentimes I've found that there is some 'problem solving' that can occur to get you back on track.
For example: I missed my flight to BOS to AUS for SXSW one year. Initially it was a 'the next available flight is in a week' situation, which clearly wasn't going to work for me. We went through some problem solving however and found a solution that totally worked. They booked me on a flight instead to DFW or SAT (can't remember which), with a layover in CLT. I went on Twitter and found a friend driving to SXSW who was going to pick me up then and take me there from the alternate nearby city. They also put me on standby for a flight from CLT to AUS. The standby totally worked out, and I got to Austin only an hour later than originally anticipated.
However, if the person at the gate/ticketing doesn't feel like helping you then frequently you're just kinda screwed. I've had dozens of situations like this where I was able to convince them to try something 'different' with me and it worked out well.
If an App helped me automatically rebook and find my options that I'd otherwise have to go beg a customer service agent for... that would be awesome. If then I'm on a connecting flight that something is going to mess up- I could probably have my plans made by the time I'm off the first plane.