Okay, we'll just say it: Gertie gives weird gifts. Dead geraniums, Gertie? We bet they're not even yours to give. They're probably your mom's.
Here's the thing, Shmooper: Bizarre or not, the geraniums that Gertie gives E.T. are a really important part of the movie. Why? Because they represent life. When Gertie first gives E.T. the flowers, they're wilted and basically dead. (See? Weird gift.) When E.T.'s alone with the geraniums not long afterward, he looks at them, hums, and they bloom back into life. So the geraniums are also a symbol of resurrection, or rising from the dead.
Later, as E.T.'s health begins to fail, the geraniums show up again. Michael's just taken a nap in E.T.'s closet when he spots them on a nearby stool. They wilt rapidly before his eyes, and he screams "No!" just before the scene cuts to E.T. crashing and dying.
Fortunately, the geraniums make another appearance, this time in the makeshift hospital after E.T. has been pronounced dead. Elliott says what he thinks is his final goodbye to E.T., only to spot the revitalized geraniums on a nearby counter as he exits. He knows a resurrected pot of geraniums means a resurrected E.T., and he's pumped.
We don't know who keeps moving the geraniums around Elliott's house (personally, we hope it's Harvey the dog), but we're glad that they do. Symbolizing life and resurrection, they're a pivotal little plant.