If this has been posted already, my apologies, but I don't have time to review 14 pages of discussion at the moment.
For anyone wanting to make simple, functional puppets on-the-cheap, I suggest checking thrift stores, yard/garage sales, and flea markets for plush characters that already have an open, working mouth that would fit your hand.
You can sometimes find interesting-looking character plush toys at these locations for well under $10. I have 5 puppets I currently use in my kids show and stand-up act which cost me less than $3 each because I found them at Goodwill or a yard sale. I bought two more last weekend at a garage sale for .50 each which I'll be modifying shortly.
Construction is very simple and easy, but first you have to decide on the location of the opening for your hand: the back or the bottom.
If you'll be using a stage curtain for performance, you might want to cut into the lower back or the bottom of the puppet so you have less arm showing and the puppet has a better aesthetic position from the viewer side. But if you'll be visible and using the puppet like a ventriloquist dummy (which is how I use most of my puppets), you'll want to cut into the plush starting just below the head where the neck begins.
Once you've made that decision, you're ready to begin.
1) Using an Exacto knife or box-cutting razor blade, make an incision (on the back or bottom of the puppet) that you feel is long enough to fit your hand inside up past the wrist. For most average-size hands, it should be about 4-6 inches from top to bottom.
You can also use scissors for better accuracy--helps in cutting a straighter, cleaner line.
2) Remove enough of the inside plush polyfil material that you can get your hand inside all the way past the wrist and get your fingers up into the mouth.
Before going any further, take some time here and go through some motions with the puppet to make sure it's comfortable on your hand/arm and that it's a good fit. You might need to make further adjustments, like increasing the size of the incision or removing a little more polyfil.
3) Take any tube sock, slide your hand/arm into it, and then insert it into the hole you made. Move it around and work the end of the sock so your fingers have a good grip on the inside of the mouth. Note the remaining length of the calf-end of the sock now sticking out of the back of the plush, mark it, and trim off the excess.
4) Sew the opening of the calf-end of the sock into the opening you cut into the back of the plush body.
And you're done!
For even simpler construction, you can go without the tube sock, but just bear in mind that you'll need to reinforce the opening with something (fabric sewn around the opening or even duck tape on the inside works well) or the opening will rip/tear open wider the more you use it.
And for a little fancier construction, find a tube sock color that's a close match for the puppet body color, or use a black tube sock to match the background curtain of a puppet stage and then you don't have to cut off the excess. And if you have more advanced sewing skills, cut a hole in the toe-end of the sock and attach the sock opening around the mouth as well so you can get a better grip on the puppet mouth and have more freedom to manipulate expression, without the mess and problem of the polyfil falling out every time you withdraw your hand.
And for a better grip on the mouth--especially if the mouth itself is a hard plastic or cardboard on the inside--I actually recommend taking a leather glove (right or left-handed, depending on your preference) and use an adhesive (like fabric glue, hot glue, or even caulk works pretty well) to secure the glove to the mouth (four fingers on the top, and the thumb on the bottom).
Only issues with gluing a glove inside is that it a) limits you to either a left or right-handed puppet, depending on which glove you chose, and b) limits your ability to manipulate the puppet's expression freely, but I have Kermit and Elmo puppets that I use with kids occasionally that the plush toy had such a heavy, solid plastic mouth that I could hardly manipulate the mouth at all if I hadn't glued a glove in.
At any rate, hope this offers some help for those on a limited budget. It's also good practice working your way up to crafting more complex puppets. I have a giant plush lobster that I already modified with a manipulative mouth and full arm for performance, but I'm giving him more character by adding working eyes and antennae.