Hotels

Google Maps is, very unhelpfully, refusing to show me the outline of the Buskerud area, so I'm not sure what villages it encompasses. Can you be more specific about your plans? When is the trip?

Although you may be going to areas not well-known to foreign visitors, the Norwegians themselves love the outdoors, so you'll be competing with them for lodgings. It's my impression that there's more demand than supply in scenic rural areas, which can lead to issues with both availability and price. However, those problems are likely most pronounced in places readily accessed by public transportation. Since you'll have a car, you won't be limited in where you stay.

I'd suggest poking around on booking.com to get an idea of what may be available on your dates in the areas you expect to travel through. Not all lodgings will be listed there, but it's an easy starting point. The next step could be zooming way in on Google Maps, jotting down the names of lodgings you see, and checking their individual websites.

Buskerud is a varied area with both large (by Norwegian standards) towns and rural areas. And while I would not worry about finding a hotel room in Drammen on short notice, I would not count on any accomodation being available without prebooking in the less populated parts.

This is an ancestry trip, the towns are Haug, Bakke and Over Eiker. Google maps are not much help in finding these towns. We plan on going in the beginning of September. Can anyone also suggest a good road map in english?

Many years ago I purchased the Avis road map, it is very compact.
I also purchased Veiatlas Norge a spiral bound atlas that has about 60 very detailed maps of Norway with other maps of specific cities.

Haug is near Hønefoss.
Bakke is a very common name.
Over Eiker, I think this is a traditional region not a city/town. Look up “ Eiker” in Wikipedia.

Norway traditionally was not made up of villages, in some cases a large farm became a small settlement in modern times.

A Kommune acted like a village, but is more on par with a township or county in the USA.