The mechanism which I wanted to build was a servo-driven vent for my final project (a greenhouse.) I originally wanted to have a swing arm or a gear mounted on the hinge being driven by a gear on the servo. Then I realized that a cheaper and more effective way of opening a hinged vent would be to have a socket attached to the vent and fishing line running to a spool being reeled in by a DC motor.

But then I realized that there was an even cheaper and more effective way of doing this: cut a mounting hole for the servo in a pane of acrylic. Glue the servo into the hole. Cut vent holes in the pane. Cut semicircular tracks for the vent to sit on, then glue them around the servo mounting, using several tracks to provide the necessary clearance for the servo to move. Then cut the vent, which is circular, has two holes which line up with the vent holes in the greenhouse pane, and sits on the tracks. The .svg files are here and here.

Advantages: the servo moves in the same plane as the vent, minimal moving parts, vent is fully supported by the tracks (for a vertically-oriented vent, semicircular tracks can be built up around its outer circumference to support it from below.)