San Borja Norte , close to the old ministry of education...but its pricey
I used to live near Av Las Artes and it is nice. Although it's not particularly cheap, I found the apartments tend to be noticeably bigger than what you find in districts like Miraflores for a similar price. And there's the new shopping centre La Rambla which replaced an old Plaza Vea with a new cinema, restaurants. In fact, round that whole area, there are lots of nice restaurants and places to go out. I specifically like Sarcletti's. San Borja Sur is a bit quieter and more residential. Towards the Surquillo end of San Borja, prices drop a bit, paticularly from Aviacion and Primavera, so if you're looking for any bargains, that's probably the area to look in.