There's an active and lively young design community in Budapest, to some extent fueled by and to some extent fueling a monthly, free-form, exuberant design fair (open-air in the warmer months) called WAMP...clothing, ceramics, glass, sometimes performance art:

http://wamp.hu/en

On Raday utca there are several galleries, one of which always has a selection of interesting design pieces...the Raday Galeria: http://www.raday-galeria.hu/?page=galeria&lang=en

On Lonyay utca (#24) there's a couple of storefronts with a fascinating trove of reproduction pewter deco tchotchkies  and other design elements, including mosaic tiles and a range of funky interior odds and ends. Owned by an Italian ex-pat, Aita Studio is not cheap but not overly expensive, and worth rummaging through.  Their contact information is on their mosaic tile website (if you can manage to cajole them into letting you see what's stashed in the basement storeroom, there's 5 times as much as they have on display...:

http://www.liquidfloor.hu/

At Károly körút 10 there is a large courtyard with easy access from the street (among the shops with signs posted are the Bon Café (a p-leasant little place for a cup of coffee), Jo-Jo and Bahia. The shops have a very eclectic range of stuff, from clothes through avant garde chess sets and jewelry, and very good prices.  Worth a wander into their various nooks and crannies.