At the Olympics, only in equestrian events do men and women compete on an equal footing, since it's about control of the horse, not strength. There used to be a truly mixed event in shooting (skeet and trap) but these were split in the 90s. (Perhaps shotgun experts will explain why this is, even though a woman won the last mixed event.) There are other mixed events -- Sailing, mixed doubles badminton, ice dancing, pairs skating, mixed doubles luge and so on, which are mixed by requiring a fixed number of men and a fixed number of women.
It would be nice to come up with some more events that can be truly mixed, but it's also pretty easy to design events mixed as defined by the rules. Mixed doubles tennis already exists. It's easy to imagine all sorts of "relay" sports with a mixed team in swimming, track and other sports. Ideally even a relay consisting of events women do particularly well at combined with events men do particularly well at. Team events are easy to mix as well, such as rowing, bobsled as well as baseball, basketball, voleyball etc. Many of the machine-based sports (car racing, motocross, etc.) don't show up in the olympics but can be, and are in some cases truly mixed.
So the idea, initially for the PR value, would be to create a games where all events are mixed in some way or another, with as many truly mixed events as possible. One imagines it should be able to get good TV ratings at least the first time for the novelty, which could turn into prizes good enough to attract top athletes to do the work of creating new teams.
In some cases, forming new teams would be trivial. 2 men and 2 women both experienced at relay racing for example, need learn very little to make a mixed relay team, and in fact mixed relays take place in amateur sport fairly regularly.
(Another interesting thought would be mixed relays of entirely different sports. One is depicted in one of the commonly running TV ads on the CBC. Some people have done relay Triathalon already, and the concept can be extended.)
I will add that I'm not trying to add more sports to the Olympics, which are already overflowing with events.