Wild-Ass Rumor: the long-awaited base ($35k) Model 3 may not have the full Autopilot hardware. Not sure how credible this is, given that the radar unit would probably be the easiest place to save the most cost... but something like this wouldn't surprise me. 
Remember, Tesla made the original base (40 kWh) Model S unable to use Superchargers and then acted shocked when it turned out there was no demand for it. If they can't make a $35k Model 3 at positive margins, it wouldn't surprise me if they did something similar with AP.
I'm not able to find a replacement cost for the Conti ARS410 CV radar unit that Tesla uses in the current AP hardware suite, but maybe @hoonable could provide some context on the rough cost involved? I'm just assuming it's the most concentrated cost of the system.
For the broader context, consider that a recently-leaked Elon Musk email suggests Tesla needs to beat $3,000 out of the Model 3 to reach the magic $35k mark. That seems to include labor though, not just BOM.
A helpful DMer points out my obvious oversight: eliminating the compute board could save more on the BOM than the radar unit, which they probably need for AEB/FCW anyway. Since Tesla is ditching DRIVE PX2 for its in-house chip, calculating the compute board cost may be tricky.
