What would the the best part of Barcelona to stay in for a first trip of 7 days? We enjoy history, cathedrals, museums,cafes, restaurants and being able to walk to many of these activities. Preferably not the noisiest or most crowded part of town.
My last visit I stayed in the eixample, towards the south end of it. Specifically near Tetuan. The whole eixample area is nice and pretty central. Near a metro stop is important for getting around. Close to both the modernista sights and the old town.
My favorite area was the El Born district - great feel - but I didn't stay there so can't speak to how hotels are.
I'd avoid Barceloneta or beach - fine areas if that is your focus, but not as central for other sight seeing. I'd also avoid too near Ramblas - noisy, over touristed and apparently full of pickpockets.
I like Eixample 'Dreta' - the area mostly to the right side of Passeig de Gracia.
http://www.barcelonas.com/dreta-de-l-eixample.html
My preference is near the Girona metro stop.
Anywhere within Passeig de Gracia - Mallorca - Passeig de Saint Joan - Gran Via de les Corts Catalanes.