It depends on how long the tear-down and new construction would take. An investor would work backwards from the prime selling season--generally early and mid-spring. And yes, weather does matter some--I'm sure Elvis (or another good Chicago agent) will correct me as to the prime time, but it's usually as soon as the weather starts turning decent.
So, let's say tear-down and new construction would take 7 months. (That's just a guestimate, and an observation based on some I've seen. And it depends on many factors.) Whatever the time period is--allowing a bit in case there are problems with permits or other unexpected issues--subtract that from the date it goes on the market. So, assuming the investor wants to put the new house on the market in February, that means he'd want to purchase it no later than July.
However, it might make sense to put it on the market earlier, and offer to lease back for a period of time. That way, you get it on the market as soon as possible--you don't want to miss your window of opportunity, either--but it gives the investor an opportunity to time the tear-down and rebuild so that it works best for him.
So, check with a Realtor familiar with Chicago regarding timing. But, in this market, considering that you indicate you're flexible regarding timing, I'd be inclined to put it on as soon as possible, offering to lease back for a period of time.
Hope that helps.