At their most basic level, I don't think much room exists for innovation in quests.
I don't think I could disagree more. Just because a lot of game developers don't
have much innovation in their quests doesn't mean that it can't
be done. I will pretty much echo Eusis's sentiments here, but the best quests are one that have some actual depth to them. They don't have to be world-changing events (in fact they probably shouldn't be, they are side material after all), but that doesn't mean they can't have well-developed characters and occasional twists and turns. Bioware sometimes gets this right, but certainly not always.
But yeah, there's nothing more boring than 'collect n units of material A' or 'kill monster X'. Developers put those sort of quests in because it's easy to do and considered safe and reliable.
Actually, I'm going to contradict myself for a second and note that the hunts in FFXII were an example of 'kill monster X' quests that I really enjoyed. But that was because there was often some research and planning involved in even finding said monsters and sometimes you also needed special tactics to be able to take them out. Furthermore, if you do enough of thoses quests you should know that there actually were
some unexpected twists and turns. So yeah, I guess even that sort of quest can be done well. Rarely is, though.
A side comment, but one thing I find really makes me more likely to engage in sidequests is some sort of in-game quest log. More and more games do this sort of thing now, but still not all of them. But it's really helpful for games that you're not necessarily playing all the time.