Accreditation is important to me at the high school level in case we wanted to go for the HS diploma through the district.
We haven't done online courses this year, but we always had software troubles--so another thing that's important to me would be bug-free software. I don't ever want to talk to tech services even once during the course!
Another thing that's important to me is that the entire course is ready to go at the beginning--no waiting for lessons, links, or videos to be uploaded.
Tutors/teachers that respond to student questions/emails in a timely manner (we'd wait weeks to get a response or grade back).
I really like the online community of AoPS that "chatter" before and after class, that have a personal blog space to work out problems, get to know each other, etc.
