Hi, Chris! Portland is just like my second home, as I have been there for many times. This is a lovely city, with pleasant weather and I really love it.
If you are looking for best yet affordable weekly motels, here are some recommendations.
1. Econo Lodge City Center (1889 SW 4th Ave.)
2. Extended Stay America - Portland - Gresham (1777 N.E. Sacramento St.)
3. Best Western Inn at the Meadows (1215 N Hayden Meadows Drive)
4. Nordic Motel (11942 Northeast Sandy Boulevard)
5. Candlewood Suites Portland Airport (11250 Northeast Holman Street)
Be sure to check the websites of these hotels and find out what amenities these hotels include and their prices. Then choose your hotel according to these aspects. If these hotels are beyond your budget, you can have a look at these cheap hotels in Portland. Then give them a ring and ask whether they offer weekly rates.
http://www.portland-hotels-oregon.com/cheap-hotels
By the way, how long do you plan to stay? If you plan t stay for a long time, more than 1 month or two months), it is better to rent an apartment. This will save you more money. Anyway, if you choose to rent an apartment in Portland, you may turn to Portland craigslist. There, you can find almost apartment for rent of any type and budget.
http://portland.craigslist.org/apa/
Good luck!