Midrange
There are several around Kumamoto and are conveniently located. The rooms are small but have everything, including free internet. The staff are courteous and helpful and tolerant of English speaking guests. Breakfast was good. It is great value. You can book on the internet but can only book a certain time in advance. Suido cho one is well placed for tourists to Kumamoto. (http://www.toyoko-inn.com/eng/)
There are a tonne of business hotels scattered around the city with most clustered either near the JR central station, or in the nightlife district in and around Sun Road, Fuji and Ginza. Some even have outlets at both. Follow the neon signs.
There are a couple of further business hotels located in and around the central bus terminal about two blocks from the nightlife district including a hotel upstairs in the bus plaza itself, and a Dormy Inn directly opposite.
Budget
Being a large city, Kumamoto has its fair share of alternative sleeping options for the non-discerning traveller willing to camp out anywhere.