I could never understand why Elliot didn't go with his best friend on adventures in space when I was a kid. And then I grew up, and realized that Elliot had matured during E.T's visit. At the beginning, he's a very typical 10-year-old, hurting from a divorce, and unable to keep that hurt to himself, even making his mom cry. His brother yells at him for that, telling him to grow up and think how other people feel for a change. When he comes back from sleeping overnight in the forest, he learns that he is just as capable of hurting his family as his father did when said father left his three kids and wife, and at the end, cannot bring himself to hurt his broken family further by leaving with the aliens in the spaceship, no matter how much he loves one of them. Ouch.
Which makes it Spielberg's rebuttal to the ending of his previous Close Encounters of the Third Kind where the father did leave his family to go gallivanting with aliens (none of whom he had formed a relationship with).
When E.T. sees Gertie for the first time she screams and he screams back. He was scared of Elliot at first, screamed, but came back and got to know Elliot. By the time he met Gertie and she screamed, he thought that was how humans greeted each other and was mimicking her facial expressions.
E.T. are the first and last letters of Elliot's name. Word of God, a trading card series to tie in with the movie says the family's last name is Taylor. Elliott could've given E.T. his initials.
Then, because Elliot relates to E.T as an outsider, the film could also be called Elliot Taylor (:The Extra-terrestrial part of the title hinting that Elliot also sees himself as an alien due to him being different and alienated after his father left home)
Why does Michael find E.T. by a river? Word Of God states that E.T. is a plant like creature (The Expanded Universe book and games touch upon this) and he could've been there to help himself survive.
E.T. is able to make the bikes fly, as well as other objects. It's possible he can't make himself fly. At the beginning of the film, why didn't he sit on a tree branch or rock and fly back to the ship?