According to the article about the flight on lostpedia
She said that windows provide a route back, but they didn't stay open long, and the window through which they were supposed to enter closed in 36 hours. She handed Jack a binder that contained all air routes that flew over the coordinates of the Island, pointing out that the intended window was directly in the flight path of Ajira Airways Flight 316. Ms. Hawking emphasized that they must take that flight if they had any hope of returning to the Island.
Essentially, the island is constantly moving and shifting around (which is what makes it so elusive). In order to get the Oceanic 6 back to the island, they needed a flight who's path would match up perfectly with where the island would be so that they can crash it onto the island to return.
There is no island rule about this having to be the flight but merely the only way to keep the island "stable" enough to save it is to have the original/remaining candidates back onto the island in order to stop it from producing the time flashes so sporadically.
Eloise is merely the one that runs The Lamp Post station which it's purpose is to detect where the island currently resides and where it will be.
Faraday's father, Widmore turns out to be the original "others" leader back in the 1950s. And since Ben banished him when he came of age, he's been desperately trying to get back to the island ever since. I'm sure anyone ejected from their longtime home that is constantly moving around the world would be pretty determined to get back at any cost.