Trivia: Sigourney Weaver was not that keen to be in the sequel to Alien, so she demanded an exceptionally high fee considering her fairly low status as a movie actress at the time and also a percentage of the box office takings. The producers were going to drop her and go ahead without her but James Cameron insisted that she was fundamental to the film and so her fee was accepted and paid.
Trivia: Early marketing research ahead of the film's release showed that a significant number of people mistook 'Aliens' for the original movie 'Alien.' To address this, the studio made sure commercials referred to the film as "Aliens: the new movie" and gave the film a logo visually distinct from the one for 'Alien.' To this day, 'Alien' and 'Aliens' are treated almost as separate brands; merchandise and tie-in media tend to evoke one or the other, rarely both.
Answer: It really was all down to James Cameron having already written the script and proving himself capable of directing with 'The Terminator.' It was just a quicker, easier, and almost certainly cheaper decision to let him direct his own script rather than get someone else, even Ridley Scott. While the producers had wanted to make an 'Alien' sequel almost immediately, at the time the head of 20th Century Fox didn't want to pursue it fearing it would be seen as an obvious cash-in and flop. When a new executive at the studio came in a couple years later, the project was put back on track, and I believe Cameron was the first to be approached to write the script.
TonyPH