Have you considered a Hostel( how old is son?? , teenager or child?)
You could look at a place called " Perfect Hotel" ,, it is a hotel , but has some shared bathrooms,, ( but also rooms with private facilities),, I think they may be in your budget range.
Your price range is very low. Convert and think in Euros.. thats about what,, 70 euros,, can you get a nice hotel in central Seattle in a decent area for 70 dollars??NOt impossible,, but very difficult. Good luck, and don't give up,, but do be careful,, if you find a place thats less then 60-70 euros,, and it claims to be central,, well, check by googling for reveiws for it,, and look a a few different sites.