I am sorry your puppy is sick. It is really hard to take their little whimpers when they are perfectly healthy pups, but a sick puppy crying is like murder to the soul of anyone who cares for animals.
As for your question, I am NOT a vet. With that in mind I will tell you two things that make pedialyte seem like a bad call:
Not a sterile solution.
The sugars might be ok, but the other ingredients which are helpful to people, might be fatal to your puppy.
A sterile solution of lactated ringers (very cheap) will be a much better choice in my non-medical opinion. It is a balanced mixture of dextrose in saline, which should also be prepared in a sterile manner.
You should just pick up the phone and call a local vet, they will likely give you the info you need at no charge, and perhaps sell you the things you need to hydrate your puppy at home. (Very cheaply too, I might add.)
Injecting under the skin might be something that comes after intubating the puppy to insert the liquids directly into the stomach. You should get ahold of your vet, or any respected veterinarian for that matter, and ask about the solution you need, and how to best administer fluids to a sick puppy.
(Notice: After reading the link below about NOT tubing pups, I would be reluctant to do so on my own. The reasons expressed by that person make perfect sense, which makes injecting sub-cutaneous fluids into the puppy look safer, when done by a vet.)
Neither of these methods should be performed by someone who has never done them before. Done wrong, either one of them will kill you puppy outright, or make the dog impossible for the vet to save.
Again, please take this advice: DO NOT do this on your own, without talking to your vet.
Here are some helpful links for you:
This guy REALLY seems to know his puppy saving techniques:
I hope you will find this useful, and maybe consider joining the site. It is free, and you get to ask the question to everybody at once as opposed to one person. Perhaps you can help people with other problems? We would love to have you.
Also, if you were a registered user, you could leave me feedback and tell me how I did. :-)
I hope your puppy makes it. Let me know how you make out: dangernerd@gmail.com if you like.
Attention: NOTHING on this site may be reproduced in any fashion whatsoever without explicit consent (in writing) of the owner of said material, unless otherwise stated on the page where the content originated. Search engines are free to index and cache our content. Users who post their account names or personal information in their questions have no expectation of privacy beyond that point for anything they disclose. Questions are otherwise considered anonymous to the general public.