No one ever takes my relocation recommendations but here they are:
Fremont - Great location, and it used to be a little quirky, too, and still kind of is. This is within a decent bike ride (3 miles, roughly) of the UW. Close to Bottleworks, Brouwers, Fremont Brewing Co., Marketime, etc. I recently lived here, and I loved it, as I could drink beer, and smoke pot on my patio in perfect peace. The apartments, and condos are on the "cheaper" side when compared to Bellevue, Downtown, Wallingford, etc.
Wallingford - Super close to Fremont, as they’re adjacent to one another, and I find the neighborhoods to be the same, in a sense, with the one key difference being that prices tend to be a bit higher, dare I say, in Wallingford for some reason. You can enjoy all of the benefits of being in Fremont, and you’re close to the U-District, but not by much depending on where you live.
The two neighborhoods above tend to have a busy feel if you live off of Fremont Ave, which is what I lived off of, or 45th. There is VERY limited housing off of 45th, so don’t expect to find much. Tuckered back off the aforementioned thoroughfares are some decent places - look there.
Downtown - This can be broken up into multiple "neighborhoods", if you live here, e.g., South Lake Union, Belltown, Dowtown, but just know that places here are fucking expensive, and the neighborhoods I mention are very, very close to one another. My wife and I pay, rougly, $2,300 per-month for a 1000 sq. ft. apartment, with two beds, two baths, and a patio. It’s a new building, and we like it. Parking is an additional $150 per-month.
With the above, I had my parking included in Fremont, and my brother and I (living with him before marriage) paid ~$1,200 per-month for a 1,000 sq. ft. apartment, which had two beds and one and a half baths, with a patio. The quality wasn’t as great as the new build, e.g., no hardwood floors, or marble counter-tops, but it definitely wasn’t a bad place. You’d be within 3 miles of Fremont, and getting over to U-District just means getting on 99 and cutting over, or jumping on I-5.
Captiol Hill - Some places are kind of shitty, some are nice. It’s really hit or miss. You can find great deals, but you just have to strike at the right time. Great location relative to the U-District, and sitting uphill from downtown.
If you need more details, and/or you seriously want to talk about where to look, let me know. This is just my opinion, and some may disagree with my assessment. Good luck, and cheers.