The California Science Center in Exposition Park in Los Angeles is perhaps most famous for its recent acquistion of the retired space shuttle Endeavour, which arrived by jet and then was trucked through the streets of LA to the delight of huge crowds. You can watch a cool time-lapse video of the move here.
They have a design/build space as well, but it being LA, I guess, it is almost entirely devoted to media.
This is a kind of unique take on design/build, which has the advantage that almost everything to do is virtual. You can produce a record, create a mix, work with samples, choreograph a light show or (again, it being LA) design rides for an amusement park.
And, like the Lawrence Hall of Science, they have another exhibit space where you can design structures to be earthquake (and flood) resistant. A California thing, I guess.