Lima is not the dangerous city it used to be during times of the Sendero and periods shortly thereafter. That said, it is still a place in which to be careful.

Miraflores is by far the best part of town, although I know nothing of the hostel you mention or the area in which it is located. You will most likely need a taxi to get around, but don't ever take a taxi that you pick up along the street, especially if you have your mother with you. Instead have the hostel arrange a taxi for you and if possible, have the taxi stay with you for the entire day. Arrange a fee for the day when picked up and this should work fine and cost very little.

As far as things to do there are many. Any good guidebook will list many of them. However, my first stop would be the central plaza (but avoid the area at night) with the old Spanish Colonial achitechture. I would also consider seeing the Museo del Oro (Gold Museum). As you will be in Maraflores, there is a lot of shoping to be had in the area in and around Larco Mar. I would also think about spending part of the day in the Indian Market were a lot of interesting native Peruvian things can be found (a lot of junk too).