Seabird is a lovely old cafe next to Aoyama Gakuin University, between Shibuya and Omotesando stations. It's been open for about 30 years and has a very homey vibe, cluttered but cozy. The menu has the old style Japanese cafe 'morning sets' (toast & coffee) and a small liquor selection.
The music is a mix of jazz styles but nothing too heavy, with small live/jam sessions on Fridays and Saturdays. Mr. and Mrs Toriumi ('Tori'鳥 = bird, 'Umi' 海 = sea) are super friendly and generous (they invited me to join them for dinner in the cafe last time I dropped by) making Seabird the kind of place you want to be a regular at. Amongst all the over-priced, soulless cafes of Omotesando, Seabird really stands out for its warmth and authenticity.
Note: there are two entrances to Seabird.
Omotesando Station Exit B1, walk straight and then turn left on the first street after Aoyama Gakuin University. Walk down and Seabird will be on the right side of the street. The whole walk should take less than 10 minutes.