If its an actual lid in that it doesn't sit on top of the compost, but on the uprights of the outside, 4mm twinwall polycarbonate? Won't keep the light out, but is insulating - you could even use 10mm twinwall, more insulation. Traditionally, people used old carpet cut to size, but that's in contact with the heap itself of course. Guess you could use that as well, to cut out the light...

You probably used general purpose plywood instead of marine plywood. Marine plywood uses a waterproof glue to bond the layers and others don't. The best wood to use for outdoor use is pressure treated timber. It's injected with a preservative under pressure and this preserves the wood from the elements for a long time. I built a bench with an arbor 7 years ago using pressure treated timber, I haven't painted or treated it since and it's still as good as the day I made it.

Interesting ! My lid is far from waterproof. Should it be ? Does this matter ?

When I had home-made wooden bins I just had a piece of old carpet over the top.

The lid is about 30 inches square and sits on top of the rim of the bin. It sits on two strips of wood to raise it slighty, as a lot of moisture is given off when the contents heat up.

Twinwall polycarbonate is a good consideration; it would need some weighting-down though.

Marine plywood - isn't this expensive - I'll look into it however.

Does it matter that my current lid is not waterproof?
Only that the layers may start separating and fall apart.

Like Bulba, I just have old carpet on the top of mine and it makes very good compost. I hadn't realised I should be keeping the light out. I just thought it was to keep it warm.

I don't think it would matter too much that the lid isn't waterproof, as long as it doesn't let in lots of water. I suggested marine ply as it wouldn't rot and fall apart. I suppose any of the above ideas would work. Polycarbonate would be a good material, if weighted down and covered with something to keep the light out to stop weeds growing.

I cover my compost with 2-3 ex-Coffee sacks, not so much to keep light out, but to keep the heat in. Eventually they rot down, but they do take some time. As stated previously, it's then topped off with the lid to keep rain off.

After the initial heating stage, the compost is dry-ish and almost 'crispy', and it requires watering, but as least it doesn't get soggy. It's easier to add water that to remove it.

I always just slung a piece of old carpet on the top of mine, never worried about keeping rain out - but, it was open at the bottom, on soil, so any excess water could drain away.