...wooden lid... what kind of glue...What kind of finish... Some kind of oil?...
A stainless lid has lots of advantages, but if you want to try wood here's what I would do. It could be a fun project.
Use a wood that is naturally rot resistant. White oak, teak, cedar, or genuine mahogany would be good. Check the
Wood Database for toxicity of the wood dust and protect yourself accordingly (especially your lungs).
For glue, Titebond III should work. West System 105 epoxy with 207 hardener is also really good for wet environments (it's expensive, but it's really good, and useful to have around for other projects).
I would definitely put a finish on the wood, to help inhibit water absorption. Don't use polyurethane, epoxy, or any other finish that forms a surface film. It will eventually crack and flake, and pieces could fall into your vessel. Not good -- those pieces are not food safe.
Mineral oil should work fine as a finish. Give the oil plenty of time to soak well into the wood, then thoroughly wipe away any excess on the surface.
If you're worried about the oil dripping into your vessel, I don't think that's much of a concern. The oil is going to soak into the wood. The oil is not going to want to mix with the H2O condensate that will form underneath the lid. In the unlikely event there are trace amounts of oil in the condensate drip, it's food safe. If there is a trace amount in the drip it's going to be so small that I don't see how it could negatively affect your brew, or you (you can drink mineral oil as a laxative).
If you don't put some type of finish on the wood, it's going to warp significantly when the bottom surface expands as it gets moist, while the top remains dry. Even finished with oil it may still warp some, but if you keep the mineral oil renewed every so often (twice a year?) it should really help in limiting warp. It's also going to be more susceptible to surface mold growth if you don't finish it.
Don't make your inner lip fit too tightly into the vessel, leave room for the wood to expand, which it will do across the grain (not along it). Use a simple construction, where all the grain is running the same direction, i.e. don't attach cross braces whose grain runs a different direction than your main panel. That will cause cracking as the main panel expands and contracts. Just edge glue the boards together, cut to shape, cut the lip, and attach a handle.
For storage, prop the lid open a bit, so any dampness left in the vessel or on the lid can escape.