Try a small brass (or s/s) slot head screw. I usually put the screw in the lathe or drill and turn or file off the countersink. Drill the sides to take small diameter brass tube, internal diameter to match the size of the unthreaded part of the screw.
Drill the box sides for the tube, put temporary spacers inside the box to hold the lid in position, and then drill through the tube into the edge of the lid.
If you have left enough of the slot in the screw head, it then becomes a simple job to fix the pivots, but it is probably better to use steel screws to thread the lid, first, if you are using brass for the finished job.
Depending on the timber used and the design, you could also use small allen head machine screws, but relieve the box sides to take the cap head.
I have used this simple and inexpensive method a number of times, without problems. Last time I visited, the Barnsley workshop uses something similar.
Mike