This has been here for a few days now, I don't know what to say. I don't think there are any hostels in Portland. I know there are very cheap hotel/motels out near the mall (about 3 mi from downtown Portland) which he can stay at and take the bus into town.
Here's the
LINK I found. Just about all of them are well under $100/night which is about as good as you can expect to do.
I'm wondering if you can't post an ad on Craigslist or something similar about renting an open room in an apartment for a few weeks and maybe paying like $200-$300? Since college has just gone back into session, there are probably PLENTY of students with available rooms that could use the money. Here's the
link to the "roomshare" section of craigslist.... Most of these people want long-term roomates, but I doubt they'd turn down the money for a short term stay while they continue looking. Many of them are asking less than $500 for a whole month.... contact the posters and see if they'll pro-rate it for your son's stay.