Julie here from Shopify Support.
Great idea! While it wouldn't be possible to modify the online checkout itself, you could offer customers the ability to enter their own price from the product page. There wouldn't necessarily be any specific themes that offer this type of feature, but it may be something a Shopify Expert can implement in your existing theme with some custom code. I also took a look at some app options for you and it looks like Negotio may be a good fit.
With Negotio, a "suggested" price will appear on your product pages, along with an option for customers to name their own price. Although this app is designed to allow customers to "negotiate" their price, it looks like you can choose to automatically approve any offers based on rules that you set; if you'd like to give customers the ability to purchase the product for $0, then you could definitely set this as the minimum price.
This app is free to install, so I recommend checking it out to see if it suits your needs. From the app's About page, you can read more about its features, view screenshots, and even access an example store so you can see the app in action. If you're ever needing assistance with this app, you can find the developer's contact information directly under the Support section of their About page. It also looks like the developers offer live chat, which can be accessed directly on their website here.
I hope this helps guide you in the right direction! Feel free to let me know what you think.
I'm one of the developers working on TipGenius, a simple tipping or donation pop-up app for your Shopify store checkout. We're the Staff Pick this week!
When a customer clicks the checkout button, TipGenius shows a customizable pop-up that let's you ask for a tip or a donation, and then sends the customer right through to payment. It's simple to setup and simple for customers to use. I think it could really help here.
Check us out at https://apps.shopify.com/tipgenius