Hi Arno,
The selection of an onsen will mostly depend whether you plan to travel by car or by train. From Omiya, you can easily take trains to Gunma-ken, Nagano-ken and Tochigi-ken. By car, you probably want to go toward Tochigi-ken, as there is a direct highway from Omiya...
Train can be pretty expensive in Japan, especially if you need to take the shinkansen - which is rather recommended for long distance. However, if you go to Tochigi-ken, there are several rather cheap trains for that journey, so it might be a good alternative.
You might want to check our
Secret Onsen website, which has an extensive list of onsen in Tochigi-ken.
Personnally, I would check first places around Okukinu, which has some of the finest onsen in the Kanto area. Among them, I would suggest to check
Hatcho Yu and
Kani Yu. There is even a
guide for that area on these forum, which includes access information and pictures. Okukinu onsen are located in the woods, and reached by a mini-trek from the bus terminal (in fact, both ryokan have a bus to pick up visitors if they do not want to walk).
To give you an idea of the price, if you travel by yourself, you probably will need to count on 10'500 yen for the return trip (including train and bus), plus the night in the choosen ryokan (check the price on their web site, as those can significantly change with the date).
Other famous onsen resorts in Tochigi-ken are Shiobara and of course Nikko. Nikko is less interesting for the onsen themselves, but has lots of places to visit. Shiobara can be reached by shinkansen followed by a short bus trip, and again you will find several great onsen there. One of my favorite is
Oami Onsen. The cost of such trip would be a little higher due to the shinkansen cost.
There are many alternatives. You also could take a bus to an onsen, including overnight buses for some destinations. But this require more planning, as you can guess.
Hope this helps,
sanji