Bryn Mawr, the 5 Claremont Colleges (~30 m outside LA), Davidson, Goucher, Haverford, Swarthmore, and Wellesley all seem to match that criterion, more or less. Then there a few LACs that are in cities (such as Colorado College, Macalester, Occidental, Reed, Rhodes, and Trinity). So not all LACs are out in the rural boondocks. Although I agree, even at the ones that are (like Grinnell or Middlebury), the schools themselves generate their own excitement.
I also think that Sarah Lawrence might not be exactly what you’re looking for. They are notoriously bad with financial aid, and Bronxville is a very rich and white area.
I think Beloit and Hampshire are great suggestions. I would also suggest Clark and Lewis & Clark.
Not knowing anything about the financial situation for you and your family, I would think Brandeis would be a good fit. Campus culture with easy access to Boston/Cambridge on the student Bran Van or the commuter train. We visited Oberlin with younger d and it is not close to Cleveland with little access except for holiday/vacations where there is transportation to the airport. Same with Vassar as it is 2 hours by Metro-North to NYC.
Wesleyan students actually do go downtown, not necessarily for entertainment (unless there’s a Wesleyan band playing), but for, fun, quirky places to eat. And, there are probably more black people per square foot in downtown Middletown than in Claremont CA, Northampton MA, Burlington VT, and Freeport ME put together.
I’m also applying to Pitzer, Oberlin, and Wesleyan! Smith and Scripps might be good options for you if you don’t mind all-womens colleges. Sarah Lawerence is probably a safety for you but it’s pretty liberal/activist and 30ish minutes outside of Manhattan.
My daughter loves Wesleyan!!! Small classes and great professors!
You should definitely check out U. of Puget Sound. It’s on the west coast, it’s in Tacoma, which is a city with a significant minority population, but it’s not right in the most urban part of it. It’s leafy and residential right around campus. The student body is almost exactly what you say you’re looking for, liberal, activist, into social justice, etc. We just returned from a visit there and they opened up the day with an impassioned speech from the very dynamic young African American woman who is president of the student body. Heck, I’ll bet if you reached out to her, she’d be happy to talk with you – she seemed super friendly. They’ve got a reputation for excellent teaching (95 and 97 out of 100 for “professors interesting” and “professors accessible” from princetonreview), and a 30 on the ACT would give you a solid shot at some merit aid from them as well.
Some of the women’s schools might work (ex. Mt. Holyoke, Smith etc.). I think Goucher giver generous merit aid.
Tufts is arguably one of, (if not the most) socially/politically active schools in the the Northeast. It is a fellow NESCAC school with Wesleyan University, but a little bigger at 5,000 undergrads. The Black population is similar to other NESCAC schools on a percentage basis, but a little larger on an absolute basis due to Tufts larger size.
International Relations and Political Science are among the most popular majors. There is a pretty rigorous language/culture distribution requirement, which is intended to promote better cross-cultural understanding.
The notion of “active citizenship” is a key part of the culture and there is actually an entire, separately endowed college (Tisch College) dedicated to community service.
Tufts is generous in terms financial aid (especially at the lower end of the income spectrum) in that it “meets full need”, but it is not as generous as Harvard (especially at the upper end of the income spectrum).
The culture is very inclusive, “politically correct”, liberal, and idealistic, but it has a bit of a pragmatic bent. There is a Republican Club on campus but the campus has a strong Democratic majority. There is a small (about 15 %) frat presence. The somewhat unique culture harks back to its founding by Universalists (who were mostly working class) in the mid 1800’s.
The interplay between Universalists (Tufts) and Unitarians (Harvard) makes for some interesting New England cultural history.
The surrounding area (Somerville/Medford) is a working class suburban city that has gentrified somewhat over the past couple of decades (after the subway was extended and Cambridge eliminated rent control). Somerville houses the second highest density of young people/artists in the country. Cambridge has the third highest density. There is a lot to do in the surrounding area of Somerville, Cambridge (1 mile away and the home of Harvard and MIT) and Boston (about 5 miles away and the home of BU and Northeastern).
Cambridge is jokingly referred to as “The Peoples Republic of Cambridge” while it is said that Somerville residents can’t afford to be quite as liberal 
Tufts recently made the national news with one of its protests, mobilizing students from colleges around Boston.
http://college.usatoday.com/2014/12/12/tufts-students-lead-indictamerica-protest/
http://somerville.wickedlocal.com/article/20141205/NEWS/141207655
When you talk about a “decent” black population, do you mean at the school, or in the surrounding area? How important a factor is this to you?
For example, while I think that Bates would be a great school for you in most ways, realize that Maine is the whitest state in the union. Lewiston actually has a burgeoning black population, but they are mostly Somali refugees, not African Americans. 