Not a bad start. It's a clever idea bringing modern technology to a culture that's stuck in the medieval ages. But your story seems a bit rushed; if you took your time, you could weave all sorts of interesting stuff into the early chapters that would only be fully revealed later on.
About the laptop 'washing up' on Neverland - On the surface it seems unlikely, but ask yourself this: How did Wendy's 'tiny dancer' get to Neverland? How did it get dis-assembled with all the parts within a few feet? How long was it there? (Fairy Mary said 'many seasons ago'.) And how are the metal parts not corroded? Seems to me this topic is worthy of its own forum.
Maybe the fairies don't actually know how 'lost things' get on the beach, and just assume they wash out of the sea.
If your story continues, consider that fairies shouldn't know what a laptop is, or even how to turn it on. Imagine taking a laptop back in time 200 years and sliding it front of someone. Unless you've actually seen a laptop PC used, you wouldn't have a clue. I think the fairies would have to do some research to understand what they have in their hands.
Same with the cellphone. Anyone who's lost their cell connection can tell you how dumb their 'smart phone' becomes. There would be no wired telephone network on Neverland, no cell towers, and no electricity to power the cell towers even if they had them. Without a microwave (or sub-sea fiber optic) connection to the Mainland, there wouldn't be any InterNet. And once again, if you've never seen anyone use a cellphone before, you'd have no clue what it was for.
Also, your opening chapter says this story starts in 1990. Cell phones from that era were pretty primitive, and laptop PCs were downright ugly - we're talking Windows 3.0 at best, or MS-DOS 4.0 at worst. You sure you don't want to move it up a decade or 2? Maybe you can get around it by stating that Neverland time does not necessarily correspond with Mainland time. So Fairy Mary could find 2016 computing devices back in 1990.
Something else I've thought about, is those tiny fairy hands would be great at working with sub-miniature electronics.
So there's some technical challenges for you. Can you write your way through them? How about if the laptop and phone belong to a girl who lost them in a move? After the fairies finish playing with the devices, they should take them back. But maybe leave a few surprises on the devices for the girl to find. How about if the fairies hide their wings and call her up using Skype? The girl wouldn't know she's actually talking to fairies! Alright, fairies jingle when they talk... so Pixie Hollow gets their first software-talent fairies to code their way past those problems.
I hope to see more chapters soon.

Please sorry about typography. All my apostrophes and quotes in the last comment are gone. Stupid FF. I cannot correct it, so, feel free to contact me, I'll explain the details.

First of all, English isnt my native language, so I can only make suggestions about general style and not about chosen words.
Fairy Mary was on a mission to find the kind of 'lost things' that would help Pixie Hollow work more efficiently. You should specify. Think about a problem. A problem that is serious enough. Otherwise the reader doesnt care.
Since you talk about technology, your story is supposed to be scientifically correct.
When that happens, it washes up onto the shores of the Neverland. […] She saw the on switch, and pressed it.
No way. There is no laptop that will still work perfectly after lying in salt water for hours. At the very least the machine would need to be repaired but most likely it stops working once and for all. Unless! Unless this is a military laptop. Google for more information. One more problem! This thing is heavy. It cannot float at all. It should lie on the bottom of the sea. I see only one explanation: the pirates have lost it while being of fairies island.
Keep in mind that a fairy weights approximately 10 grams and her strength cannot be exceeding her weight thousand times or so. How strong is a fairy is a great topic for my new article (I have already calculated height and weight). The relationship between weight, height and muscle strength is certainly not linear and right now I can only give you an order of magnitude based on study about measuring the strength of mice. Long story short: up to 0.2 kilograms of force.
The laptop instantly shrunk to fairy size on contact, and boomeranged back to her in the process.
Sorry, Jacob, but you are a bit lazy and you are cheating. The fairies should have taken the big broken device apart, study it, understand the principles of electronics and informatics and after several years build their own computers: first using lamps and then transistors.
Moreover! Show us the confrontation between conservative fairies (or religious fanatics—even better) who want to burn the device and progressive younger fairies who want to study and create.
Right now this is a very poor start. If you are writing about informatics, please show us that you understand the subject.
Last but not least: if you just copy human devices and give them to fairies, you will ruin the aesthetics at once! Keep in mind that we all love fairies mostly because of aesthetics. Dont kill the beauty.
Anyway, Im pleased to see a new colleague! Finally, someone writes anything close to popular science and science fiction in fairies world. Good luck!

Do I finally have a competitor? Because I explain how science works using the Tinkerbells universe as the stage.
Well, not really. It looks like your Pixie Hollow is completely different. Are you sure, this kind of modernization wont destroy the esthetics we like the movies for?