Weymouth and Portland is a holiday area, which means that there are tons of B&Bs.  During the summer holidays you might find it more difficult to find somewhere. 

The following are options for budget accommodation: 

​YMCA Portland:
http://www.portlandymca.org.uk/accomodation/
01305 823761
Prices there are from £16 a night. 
 
YHA Portland:
http://www.yha.org.uk/hostel/portland?utm_source=google&utm_medium=maps&utm_campaign=google-places
0800 0191 700
Beds from £15 and private rooms from £29.
 
Bunkhouse hostel Weymouth:
http://www.bunkhouseplus.co.uk/
01305 775228
Double ensuite room from £35
Dorm room from £15