Return rental car to Gatwick or Heathrow?

Return the car in Heathrow unless that costs significantly more
The bus between Gatwick and Heathrow is slow and often gets delayed in traffic (the infamous M25). Especially as you plan to overnight at Heathrow.

Thank you for your reply. We are concerned about driving the car into Heathrow since we are not accustomed to driving on the "wrong" side of the road and have heard horror stories of the traffic into Heathrow from the south.

You don't say which direction you will be coming from, whether Gatwick will be nearer or Heathrow.
Since you will be by this time have been driving on the correct side of the road for a week, I don't think you will have issues. Make sure know where you are going. Heathrow terminals are not all in the same place, and car hire return locations could be anywhere around the airport.

Coming from the Southeast might mean Gatwick is closer and you'll avoid the M25. Driving to Heathrow isn't difficult, it's really quite easy particularly from the south but I'm assuming that the horror stories you've heard involve the M25. Whilst it is the bane of all who drive on it there's nothing particularly horrible about it other than the sheer volume of traffic and the tendency for reduced speeds. Typically my drive to Heathrow from Portsmouth involves a speedy, hassle free jaunt along the A3 before turning onto the M25 and hitting the traffic, from that point on it's usually a slow but undemanding plod until the turnoff to the airport. I see no value in dropping off at Gatiwck and then making your way to Heathrow which will still entail a journey on the M25 only someone else is driving and you're paying them. As you'll be overnighting you won't have the additional pressure of whether you're going to get to the airport in time so I would just suck up the M25 and get there when you get there.

For convenience, it makes sense to drop off at Heathrow. Are their additional charges to drop off at Heathrow instead of Gatwick vs. train or bus between the two? We drove in England for the first time in September and my first day of driving was a bit stressful when going through intersections and traffic circles, but I figured it out quickly. Highway driving was the easiest because all traffic is flowing in the same direction and I wasn't worried about turning into the wrong lane. We drove to Gatwick from the Oxford area on a Saturday afternoon and traffic around Heathrow was horrible. Not frightening, just slow.