Cookies really isn't likely what you want, they're *very* dependent on the user side, and a lot of users anymore kill cookies by default, either completely or at browser close (meaning the cookie would be lost within your week timeframe). Also, with cookies, the image shown to a given user would differ from one to the next, so any user to user interaction about it would be rather useless, as they could be talking to each other about any of the images and never realize it (people assume consistency).
You could, though, use some server side code (I could write up something to do the job in php fairly easily) to detect what day/week it is and pull up another image. The work of it would all be transparent to the user, who'd see it as though you simply replaced the old weekly feature with the new. If you have anything other than php (but still server side), I'm rather helpless, but may be able to point you in the right direction.
As for doing it only with JS, my best idea is just to set up a pre-written array of image filenames, and have it pull the user's date, subtract an initial date (converted to seconds or days from some given epoch), convert that to weeks from the initial date, and then use that as your array index to pick what image you should be using. Getting it truly "random" but consistent across all users per week ... might be possible, but the how (without server side coding) isn't coming to me. Oh, and the above *is* limited in consistency by the user's date/time being accurate, but that's their fault, at that point, not yours.