How can I rent a car in Seattle or Vancouver when I only know the date of arrival? I want to be flexible on the return date (+/- 1 week or so).
If this is possible, what are the options to make it cheaper?
You can always (or almost always) extend a car rental. Generally this is done by just calling the rental agency 24 hours before the scheduled drop-off time to request additional time.
Naturally, it happens automatically, as well, if you don't return the vehicle as scheduled, although it would very possibly incur penalties to do so. And if you wait too long (and don't return phone calls), it would likely incur an arrest warrant as well. :)
Your best bet would simply be to talk to the rental agencies you are considering, and find out what their policy is on extending a rental. Many will have this information on their web site in the fine print as well, but a phone call is likely easier than reading through all of that fine print.