The simplest method would be to price a product at $1 and allow the customer to enter in a quantity of dollars.
The more complicated method would require some custom programming where you'd have to activate a product as that type in the admin, then modify the store to know that the person can enter their own price for that product.
Development has been moved to github. Please add all issues or requested code fixes under the issues tab at the following website:
@FlyingKites: The whole error handling portion has been updated and pushed to GitHub (you'll need to download it from the link above). It now handles all error responses correctly. Please test it and let me know if there are any problems with it. If it's fine I'll update the contribution.
@jrcreasy: I honestly have no idea. Try disabling sending emails in your admin configuration settings and see if that fixes the problem. Also try using a different payment module like COD. More than likely the problem is further down in your checkout flow.
@AM/PM-Girl: Yes, that bug fix I showed you needs to be applied whether you use this module or not.