This young teenager's time in the novel is short but crucial. She provides the foundation from which the gestalt begins to grow. In other words, pay attention, yo.
Evelyn's outstanding trait is innocence, and once that is understood, the meaning behind her actions in the novel and her sci-fi superpower become clear.
According to Baby, an 'innocent' is a grown person who can talk like babies (1.29.21-22). That's what Evelyn is doing when she sends out the call to Lone, talking like the babies whose silent speech Lone hears in the very first section of the book. You know, all those talking babies in that crazy italicized section (1.1.11).
So what makes Evelyn innocent enough to be able to send out the call? The novel strongly implies it's because she's so in touch with nature and not abstract knowledge.
She's compared to a "speckled trout" (1.2.6) and it is "the part of spring where [...] all the world's in a rush to be beautiful" (1.2.8) that inspires her to weep and unhook her collar in violation of Mr. Kew's philosophy. The next moment we encounter her, she's sitting on the window seat singing about the beauty of nature and of rain, wind, and leaves touching her. As she's dying, she tells Alicia that she sees "The love, with the sun on its body!" (1.8.24).
Evelyn doesn't read, even though she's fifteen at this time. "She had not been taught to read, but only to listen and obey. She had never learned to seek, but only to accept" (1.2.6). You know, all that book learnin' can get in the way of sweet, innocent thoughts.
It's the beauty of nature that compels her to seek out touching and nakedness. Evelyn's innocence leads her to find the gestalt, which offers her the touching, honesty, and sense of belonging she craves.
You know those two lines about how it rained for a day and a night and for half the next day? They confused us a few times, so here's what's going on with that chronology.
Those two lines can be thought of as clues to make sure readers understand when Evelyn initially sent the call. Lone had already reached the brook on the other side of the picket fence (1.5.1) before he noticed the rain (1.5.10). Meanwhile, that span of rain started (1.4.1) before Evelyn began singing (1.4.3). Do some Sherlock Holmes action in your brain and you can deduce that she initially calls for Lone when she's unhooking her collar and weeping at the beauty of nature (1.2.7-9). She just doesn't notify the reader that she's sending a call because she doesn't know it herself.
So that's her sci-fi superpower! It brings Lone to her and they merge. Not just metaphorically merge, the book indicates. They actually merge, even though the book says they're barely touching and definitely not having sex:
Their silent radiations reached out to each other, mixed and mingled and meshed. Silently they lived in each other [...] their inner selves merged between them. (1.7.3-5)
Later, Lone describes the moment as a second where "there was this other, and himself, and a flow between them without guards or screens or barriers—no language to stumble over, no ideas to understand, nothing at all but a merging" (1.29.17)
Evelyn's death comes after she attempts to share her desire to be touched with Alicia, after she sends the call but before Lone finds her. Alicia reports this violation of their father's philosophy to Mr. Kew, and he goes out with the whip to kill her. Yeah, that's a totally reasonable reaction.
When Evelyn dies, she's described as 'wise.' That's when she tells Alicia love is a good thing and asks her to promise to do something for her, even though her older sister won't want to do it. Alicia promises. Evelyn tells her to "Make a wind with running and moving," that is to say, dance, naked in the sun (1.8.24).
Evelyn's wisdom was feeling the beauty of nature so much that she sent out the call that got the gestalt going. She represents the opposite of Mr. Kew's philosophy, despite his intentions and despite his killing her. As Lone later thinks, he had "made something" with her (1.29.28). That thing is the gestalt. It's aliiiiive!