Bah, they should extend the season with a special "test" race, make it a cash prize. I don't know how much the teams make per race or it costs to run each race so the prize needs to be appropriate, but lets just say, 1mil in prizes for qualifying, 500k 1st, 400k 2nd, etc, etc. Then 500k for the race win, 400k 2nd, etc, etc, then 1st in qualifying starts from the back. I'd also throw out the back 4, they are just too slow and would be more likely to cause a crash than make the race better in any way.
Pick the right track and we'll have an overtaking fest of crazyness. No tyre rules, or, everyone has to do 3 stops on whatever tyre they want and go balls out start to finish.
The biggest issue at the moment in F1 in regards to overtaking is, the cars are so aerodynamically active, they move the air around so significantly that as soon as you get in that window you're losing efficiency and it becomes insanely hard to get past the guy infront. You can be half a second a lap faster up to 1-2 seconds behind, then be unable to get any closer.
Might not fix it for the season, but maybe we can have a few "event" type races throughout the year, they'd shake things up, certainly make things more interesting and we'd certainly get a lot of overtaking.