Here is a list of some budget hotels/guest houses that our volunteers have recommended.
Please feel free to recommend some more hotels to this discussion topic.

HCM City (Saigon) District 1 (Or close to D1)

Thien Vu Hotel (15 mins away from Go Vap orphanage & 15 mins from District 1)
Approx $16/night
http://www.thienvuhotel.vn

Kim Hotel on Bui Vien street in district 1.
Approx $18-22/night. 
www.kimhotel.com

Saigon Backpackers Hostel
$10/night
http://www.agoda.com/asia/vietnam/ho_chi_minh_city/saigon_backpackers_hostel.html


96 Bui Vien Guesthouse- Saigon
Approx $13- $17/room
http://www.agoda.com/asia/vietnam/ho_chi_minh_city/96_bui_vien_guesthouse.html

Saigon Mini Hotel
Approx $23/night
http://saigonminihotel.com/smh/

Xuan Mai Hotel
Approx $23/night
http://www.tripadvisor.com/Hotel_Review-g293925-d1379492-Reviews-Xuan_Mai_Hotel-Ho_Chi_Minh_City.html

Dai Nam
Approx $40/night
http://www.agoda.com/asia/vietnam/ho_chi_minh_city/dai_nam_hotel.html

Lavender Hotel
Approx $50+/night
http://www.agoda.com/asia/vietnam/ho_chi_minh_city/lavender_hotel.html


For a range of hotels you can find on here & please double check on trip advisor.
http://hotels-in-vietnam.com/ho_chi_minh_saigon_hotels.html

Hanoi

Tung Trang Hotel
Approx $12/night
www.tungtranghotel.com

Hanoi Gecko Hotel
From $30/night
http://hanoigeckohotel.com/


Nha Trang

Khanh Duy Hotel
Approx $15-$19/room for 2 people
http://www.khanhduyhotelnhatrang.com.vn/

Golden Lotus Hotel
From $25/night
http://www.hotelgoldenlotus.com/

The Light Hotel
Approx $60/night
http://www.thelighthotel.com.vn/

The Novotel
Approx $66/night
http://www.novotel.com/gb/hotel-6033-novotel-nha-trang/index.shtml