If museums are on your list, you might want to check out the upcoming exhibitions at Jincheng Minfang Kangdao Museum in Kinmen Island. If you want to see more of the museums in Jincheng, you can easily walk to Kinmen Military Headquarters of the Qing Dynasty.
Hostels can be a cheaper alternative to Kinmen County hotels, as you can stay in a shared dormitory. There are 6 hostels in Kinmen County. Happy Tree Hostel offers free in-room WiFi and free parking. Horseshoe Crab Cottage - Hostel has free in-room WiFi and free parking. Take a Break B&B - Hostel is another low-cost option.
What are the best cheap things to do in Kinmen County?
Sightseeing doesn't have to be expensive. You if want to get outdoors, try a visit to Kinmen National Park or make a stop at attractions like Chastity Arch for Qiu Liang-gong's Mother. Be certain to include a visit to Jiangongyu Islet.