Well, I didn't stop at mini-painting, but also found a great, large box for HeroQuest which is large enough for all the original materials, plus a lot of new miniatures. The original plan was to build one myself, but after checking the wood boards for sale in the local DIY store, I changed my mind (the wood looked either too thin or too heavy).
So I started checking ebay and amazon etc. and finally found a box that was large enough to fit the HeroQuest board (I would have liked more a smaller but taller box, but to fit the original board inside, I had to go for large and flat... Perhaps in the future I'll build my own board, to fold in 4 or 6 instead of 2...). Anyway: here is the box (closed... not much to see).
As you can see from the link, the key was to look for a wine crate, large enough to hold 6 bottles in a single row. All the other wooden boxes (for jewels or decoupage) were too small for the HeroQuest board. Wine crates, though: large enough. Before you ask - this came empty (no wine inside) but I guess you could also find one which combines 6 nice bottles you'd enjoy, and keep the box for HeroQuest!
And there is plenty of space for the board, the terrible, large (too large) board.
The other two foam trays in the picture contain all the original miniatures (on the left) and many added miniatures (on the right) like alternative heroes.
And with the foam trays inside, here it is! I promised myself this summer that I would have had a painted HeroQuest set in a nice, organized box in time for christmas... and I made it, plus additional miniatures, and with the lockdown, all chrismas gaming is cancelled :-) I guess we'll play next year...
There is still room for improvements: I am thinking of the armory summary on the inside of the lid, plus maybe a rules summary or maybe something that would hold the Quest Book open at the proper page with the map... and of course, fill the last foam tray with another ten miniatures.
And because someone asked, here is the link for the foam trays: https://www.ebay.it/itm/KR-Multicase-New-N4-Trays-GW-Size-Pack-of-4/351211096900