Nowadays, buying souvenirs in Lima has become quite an easy task. Actually, your biggest problem would be deciding what to choose, since there are so many good options; and now many of these options have moved to the very same neighborhood, in the heart of Miraflores. In fact, many of these are located in the same block, very close to the Ovalo de Miraflores, probably very close to your hotel. 
Artesanias Miraflores — Photo courtesy of Isabel Guerra
Traditionally, souvenirs and handicrafts in Peru are sold in small shops or street fairs. These commercial galleries follow the same concept, and almost all of them are are located next door one to another, offering a whole world of stuff from all over Peru. You will find Artesanias Miraflores and Kaypi Artesanias on block 55, followed by the Gran Chimu Handicraft Market, Artesania Señor de Sipan, Gran Chavin, the Cuzco Market, La Portada del Sol and the Inka Market on block 54.
Chimu — Photo courtesy of Isabel Guerra 
Another gallery — Photo courtesy of Isabel Guera
These and many other shops are all located between blocks 52 to 55 of Petit Thouars avenue. Silver jewellery, traditional bijouterie, alpaca and cotton clothes, leather work, art pieces, carved wood items, accessories, handicraft, typical souvenirs, you name it; it is likely that you will find it here, so this is a very convenient option. 
Some handicrafts — Photo courtesy of Isabel Guerra
And in these places bargaining is welcome. However, be realistic and do not expect spectacular discounts. If you buy a great number of items you might get a significant discount: otherwise, it probably won’t be more than S/5. Soles.
All these handicrafts markets are located near the corner of Ricardo Palma and Petit Thouars Avenues, only one block away from Ovalo de Miraflores, close enough to most hotels and businesses in the area. 
Cuzco Market — Photo courtesy of Isabel Guerra
And as a plus, there are at least three large and visible ATMs there, in case you get carried away with your shopping spree and need to get some more cash.