Where is the best part Lisbon to stay in?

Near Rossio Square, have I been told.

"Best" in what sense?

Barrio Alto, but ideally close to the Rossio Square. When I was there, I stayed at the Destination Beach Hostel by the waterfront. It was quiet, but walking distance to Barrio Alto and the Square.

http://www.flickr.com/photos/youngadventure/sets/72157632246049955/

Rossio is the main square in the Baixa district and anywhere within walking distance of this is generally a good place to be based in. Bairo Alto can get very noisy on Friday/Saturday nights, while the narrow streets of Alfama late at night have had issues with muggings etc

Lisbon has very handy public transports ranging from metro to buses and taxis aren't that expensive either. So I would recommend choosing places where you can get good transport links:
- Saldanha, Campo Grande, Areeiro - central, safe, areas where you can easily get a metro to Rossio/Bairro Alto or parque das nações, a train to Sintra or a bus to Belém.
I've stayed here with my partner more than once and found it really handy:
https://pt.airbnb.com/rooms/706651