Shambala coffee bar is a great basement where you can shelter from the Hungarian rain, have a drink and enjoy some unpretentious gourmet food. The Euro-Asian decor combines conventional wooden tables and Japanese-style tatami. In the centre is a stone pond with goldfish, creating a relaxing atmosphere. For me, the best things are the coffees and teas (there are something like 90 varieties) and the desserts. Try a Turkish tea and orange yogurt cake, walnuts and honey on toast, or brie with rocket and sun dried tomatoes. All delicious, and amazing prices: about 8 euros for all three. Also, another bonus is they have free Wifi (without having to enter keys). The downside: the service, the waiters are very young and inexperienced. Slow to serve and sometimes they get the order and/or client wrong. Oh well...
Lumen Kavezo (also known as the Lumen Cafe) is a popular cafe in Mikszáth tér, a small square in the heart of Budapest's District 8 that is ground zero for the cultural and artistic renaissance currently underway in this once-dangerous neighborhood. If you'r out and about exploring District VII (something I highly recommend doing on a free afternoon in the city), it's the best place around to get a sense of what's happening in modern, alternative Budapest.
Aside from being your usual artsy European cafe, with good coffee and homemade baked goods, Lumen Kavezo also holds regular photography exhibits showcasing the vanguard of Hungarian and international photography. If you're at all interested in photography, it's worth a stop just to see what's on display. If coffee isn't your thing, they also have a selection of light meals an Hungarian craft beers that are worth a try.
All in all, a cool place to check out and get your caffeine fix in the 8th District.