I just loaded my Ansco Shur-Shot Jr. this morning, so its interior is fresh in my mind. Many other cardboard box cameras have the same overall construction: a wood board inside with the lens and shutter mounted to it, and the film loads on a pull-out cone that protects the film coming from and going to the spools from reflections and scatter from the gate during exposure.
For those, it would be simple to mount your threaded insert into piece of 1x2 (or even a little smaller wood) and just screw that into the board where the lens mounts. Measure carefully before drilling, and do predrill -- wouldn't want to split the wood. If you want mounts on both bottom and side (to allow portrait and landscape orientation), you could also mount a 1/4-20 acorn nut to the inside of the actual box -- on the bottom, at least, there's plenty of clearance to get the film roll past, but on the broader side you might have to use a plain nut and glue a piece of something opaque over it to block light and still clear the inner "cone". Probably prudent to get a sheet of 1/16 or so model airplane plywood to go on the inside of the box to reinforce the mounting. I recommend JB Weld for the adhesive; it's opaque, and far stronger than you need. BTW. this method would also be almost invisible from the outside.