In the Neopian Plaza, there is something called The Wishing Well. The Wishing Well sounds like a generic name, but it actually is a REAL wishing well. It grants 40 wishes every day in groups of 4 (so 10 wishes in 1 group). Basically, all you do when you get to the page is to type in the item you want to wish for and then donate some money.
According to others, the rarity of the item has to be fairly low, maybe around the 90s. So don’t throw money into the well and expect a Royal Paintbrush or a Draik Egg. The usual items that are asked for are Snorkle Snouts (for the Kadoatery) or a Bag of Peanuts (used on an Elephante) to get the avatar. Also, the amount of money you donate does not increase your chances of being picked. If you donate 1 NP or 100NP or even 1,000NP, it doesn’t change your chances. Everything is random.
The Wishing Well:
It has also been confirmed that you can receive an avatar if your wish gets granted.
Here’s a picture of that: