Not sure about Manly - I don't think any of the accommodation there could be classed as "cheap" (but depends on your definition of cheap!)
If you don't have your heart set on Manly, a bit further north is the YHA at Collaroy Beach - they have family rooms with private bathroom & cooking facilities YHA Collaroy
Might be worth a look. I haven't stayed there, but grew up in the area. You also don't have to be a member to stay at a YHA - you just pay a 'non-member' rate (check this - it's been a while since I stayed at a YHA).
The Collaroy YHA is across the road from the beach, a bus ride to the city/Manly (buses run fairly frequently on Pittwater Road, the main road through Collaroy & there are expresses that get to the city in about 45 mins, or go to Manly & catch a ferry), there's a movie theatre, various restaurants & cafes, bus ride to Warringah Mall (major shopping centre, if that's your thing), "Home & Away" bus tours to Whale Beach (for the fans, but probably a good way to see the sights on the peninsula as well).