There are several decent places to stay in or near the U district:
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Graduate Hotel (formerly Deca Hotel) - an art deco highrise hotel with great views from the upper floors.
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Residence Inn - a new hotel located very close to the Deca. Typical Residence Inn stuff - units with kitchens, full breakfast included, etc.
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Silver Cloud/University - an excellent local chain; the University branch is located on 25th Avenue NE, close to the terrific University Village shopping center; there are numerous restaurants close by.
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Watertown - part of the regional Pineapple chain - excellent hotel with a great breakfast/lunch café (Portage Bay Café) on the premises. Their sister property, the
University Inn, is a block away.
As for Mount Rainier, I'd second the Packwood recommendation. In August one can expect epic crowds as the wildflowers will be emerging.