I’ve always suspected this is to accommodate for the last District line train to Richmond. However I’ve never checked the timetable to confirm when that is. This potentially allows users to reach this destination if they miss the last train at Hammersmith.

Similar to how the last westbound central line train to Ealing Broadway holds at White City to allow the final westbound train of the night (towards West Ruislip) to transfer the last stragglers from the city

They just stop late (and early) for the more mundane reason of 'because they can', the service is less frequent at off-peak times so there's enough time to stop at Turnham Green before the train behind catches up.

It's such a busy station for being a local station (out of the TfL station types, the least-used), and the District Line beyond this is often quite crowded at rush hour -- it's rare that I get a seat starting from Turnham Green at rush hour.

Having the Piccadilly stopping here during the day would allow for better passenger distribution -- I've often found that one or the other line is brimming with passengers even when the other is half-empty.