39cm x 39cm. hmm is it one of those nice 15" cubes? i like those!
What do you want in the lid? just a fluorescent? do you have one? the type with a separate starter unit and waterproof caps are best.
One method thats worth a try for easy and on a budget, correx, its that corregated plastic used on 'for sale' signs. you can get if off ebay in UK very cheap. Cut with scissors or a craft blade and it bends in one direction not the other which can be used to your advantage for an angle-fronted lid. You would cut something more solid for the 2 sides, wood maybe, cut this to the desired profiles of the lid, draw round it for the second. Wrap the correx (or metal) around the ends and glue down. then silicone 2 pieces of correx to the inside the same shape of the end profiles, silicone seal the edges inside so correx is the only thing exposed to the humidity.
just an idea!
depends how handy you are, budget etc.
heres some quick 3D model renders as an example. After rendering i though of wardrobe clothes rail brackets inside for the fluorescent tube holder, they are plastic and U-shaped so perfect!
anyway, not detailed plans but hope they help!