I found a couple of apps in the Shopify App Store that can help make this possible. Check these out:
Both of these apps add an option on your product pages for your customer to make an offer as to the price they’d like to pay for your products. Please contact the app developer if you have any specific questions about how the app works, pricing, or need a hand setting it up. And for additional information about working with third party apps, please visit this Help Doc.
All the best!