Elizabeth Anderson is a lieutenant for John Connor in the future as well as her brother, William Anderson. In 2004 (new timeline, 1997 in old timeline), a T-X was sent in the past to kill some of John Connor's lieutenants. It gunned down her brother in their house during what appeared to be a party and then killed her while she was upstairs playing a video game. Skynet sent the T-X back in time hoping it will be able to kill the 23 people on its hit list (including Anderson family) because killing John Connor was impossible: He was living off the grid at this time and could not be located.