I strongly agree with your third paragraph. Apart from the obvious things like schematics, BOMs etc there are the less tangible things like, as you say, why you chose a particular component and just as important, why you didn't choose a perhaps more obvious component.
Now I might be showing my age here but I still like to use a hardback ruled log book to record my thought processes & design decisions - even the wrong ones. If someone in the future tries to replace a component with a more 'suitable' one or moves a track on the PCB, my notes might tell them that I've already been there and burnt my fingers (perhaps literally!).
I always number the pages and allow a few pages at the front as a table of contents. You can also document such things as calculations of power dissipation, tolerances, timings etc. (this habit comes from my days in the aerospace industry where keeping a log book was mandatory). Of course you could always put this information in a WP document but I'll stick to paper!
Circuit descriptions might also be appropriate where unusual (especially analogue) circuits are concerned. I would treat these like software comments to document any unobvious circuit or components functions. Schematics, like software should be 'self-documenting' as far as possible but sometimes this isn't enough.
A more up-to-date alternative, especially in an educational environment, might be to have a project website. This could be arranged as a collection of blogs for each discipline - hardware design, pcb layout, software etc. The blog nature would allow contributors to show their thought flow and document the ongoing progress of the project whilst other pages could be more formal (progress Gantt charts, test results etc). You could even add meeting minutes and action lists.
Hyperlinks make cross-referencing easy and now we have MathJax so even design equations are simple to insert.