I quite like Sylvius' suggestion of getting rid of the main/side quest divison. Not certain how well it would work, but I'd be interested to try it.
But assuming they don't do that, the one thing I really want is for the sidequests, regardless of the exact content of them, to make sense in the context of the main plot.
ME1 for example, did this extremely badly. There was simply no justification for your character to be doing many of the sidequests given the story. The central plot is telling us that there is a concrete and immediate threat to the galaxy, and you always have a direct lead on that threat. So why would Shepard ever do any of the sidequests (or exploration for that matter) on non-plot worlds? There simply isn't the time to do them. The ones that you can do while advancing the plot - say, for example, that guy on Feros who asks you to pick up some data from the Exo-Geni building, is fine, as you're in there anyway, and it'll only take a couple of minutes - but anything involving diverting from trying to stop Saren (with the possible exception of the Geth in Armstrong, as you could believe they were connected) makes no sense what so ever.
DA:I suffered from this too. There certainly was some need to divert from the main story, as the Inqusition needed to gain power, and some of the side content was clearly directly linked to Cory, so completing it would interfere with his plans, but at the same time, you again always have clear informaiton on what the massive threat to the world is up to and the story implies there are serious time constraints on you (ones the game utterly fails to implement, much to my dismay), so in the context of the story, the bulk of the content makes no sense for the Inquisitor to be doing.
BW's more linear, less "open world" games work a lot better, as the bulk of the side content is stuff you can do while advancing the plot. You don't have to go out of your way to do it. And, of course, to be fair, this isn't by any means just a Bioware failing. Most open world games suffer from this exact problem.
As for how I would fix it, well, the obvious answer is to put periods in the main plot where you don't know exactly what you have to do to advance it. To have times when you have to go out exploring, looking for clues, doing sidequests in the hope of finding information (and conversely, when you do have an immediate lead, you should have to follow it up rather than pissing about helping people with their life problems).