Irrigation system: I shouldn't need to punch buttons on a controller box outside in the rain.
Anything with a clock, e.g. microwave, thermostat: It should get the right time itself.
Pool heater: I should be able to control it from the house and check the temperature.
Barbecue: It should let me know if it was left on (like happened yesterday).
Freezer: It should let me know if the door is ajar, before everything thaws.
Alarm system: monitoring and control.
Stereo: should be able to control from my phone.
Washing machine: notification if I left wet clothes in it, or if it goes off balance and stops.
These are mostly available now, but not in an easy-to-use way. Home automation seems to be like the home computer industry in the 1970s: you have to really want to do it, you need to be a bit of a hacker, it costs a fair bit, and what you end up with is pretty primitive. I think there's a huge market for someone to solve home automation. What I want is when I buy an irrigation controller from Home Depot (for example), it "just works" as part of a home system.
Edit: A couple people asked why connect to the internet? I should be able to do these things remotely, e.g. from the office or my phone.
a_c_s mentions that the time tinkering could outweigh the benefit. That's kind of what I'm getting at with my 1970s home computer analogy - you could do e.g. word processing back then, but it wasn't worth the difficulty for most people. Now non-technical people can buy a computer at Best Buy and easily do word processing. Likewise, home automation needs to become something that is built in to products and "just works" by default, rather than something for hackers.