My favorite place to stay is at the Nile River Explorers/Backpackers Hostel. The hostel has a main accommodation near the town of Jinja and also a camp right at Bujagali falls where you can see the rafting. The people that work for NRE are really nice and always willing to help you find something you might need while staying in Jinja. If you don't like rafting or kayaking, check out bike rentals from NRE. You can even get a guide to take you out to the Mabira forest.
Around the corner from the NRE hostel is a place called Three Friends which has a lot of great food, drinks and even has ice cream! (pretty hard to find in Uganda) Just ask one of the guys where it's at and they'll point you in the right direction.
There is also the Adrift camp which most people stay at who book with Adrift while in Kampala. This camp has mostly tourists and can sometimes be a bit quiet when there are only a few rafters around. I prefer NRE because many of the locals hang out there and know the good places to go around Jinja.
In addition to these two options there are many smaller guest houses sprinkled around Jinja which offer a more low-key spot for people who may be traveling as a couple and don't want to stay with a younger backpacking crowd.
Try King Fisher Safaris Resort or Bridgeway guest house