I haven't done it very often, so my experience might not be typical.. that being said though:

I always get the offer to come up about midway through the check-in process. For international flights, I type in all of my passport info, then, press the "seats" button. It's once I do that that I get redirected to a page that makes the offer.

In my experience, the price that it offers at OLCI doesn't change at all throughout the time you are allowed to do so. On a recent flight, I did the first half of OLCI about 5 times to see if the offer would change, but it stayed the same.... but then was lower at check-in and at the gate.