What is the best DIY way to mount a small dome camera inside a car? this should be watching the back seats only where children or pets usually are. This is the type of camera I am thinking about: I am not talking about wiring and connections since this could be battery operated and completely independent from the car wiring, only a quick and easy way to attach it somewhere so that it has a clear and stable look of the back seats
One good option which won't require are drilling or other permanent changes to the interior fabric of the vehicle would do to use a headrest camera mount. This is a device which attaches to the headrest supports of the driver or front passenger seat and presents a standard camera mount.
These mounts are specifically designed for the exact application you describe and many variants from a number of manufacturers are available.
There are several dashcams on the market which have two cameras, and mount over the rear-view mirror. One of the cameras is turned around to cover the interior of the car. Taxi drivers use them a lot. Just because you happen to have a battery operated camera doesn't mean it's the best choice for the job.
If your back seat has grab handles then that's the place to attach the camera, they handles are above the door so will have a decent field of view. The handle is securely attached to the car so not going anywhere in an accident.
The primary concern for attaching the camera should be safety, you don't want it coming loose and smacking someone in the head. You could bend some brackets around and screw them together for a good secure, semi-permanent attachment. If you want a temporary attachment then you could use double sided velcro strip, the stuff's amazingly strong.
However, your choice of camera is inherently dangerous because the housing of home dome cameras is not designed to hold up in a collision. In an accident the camera might break free no matter how secure your attachment point. If it were me I'd get a gopro style camera and use the handlebar adapter on one of the rear seat's grab handle.