What do we call a price that is chosen by a customer?

I'm a native French speaker.

In recent years, there has been a new concept of price in French that is "prix libre". A literal translation of "prix libre" is "free price".

This concept means that a seller proposes that the customer has to decide what amount of money he is willing to give for a given good or service. It especially applies to some cultural events. It implies that you may purchase something with a "prix libre" for free (0$/€/...), but the seller encourages the customer to pay something. It is not really seen as a "tip".

My goal is to put this expression on a website where we propose to download a document at "prix libre".

The phrase "Pay what you can" is used.

Pay what you can (PWYC) is a non-profit or for-profit business model which does not depend on set prices for its goods, but instead asks customers to pay what they feel the product or service is worth to them.[...]

"Pay what you want" is sometimes used synonymously, but "pay what you can" is often more oriented to charity or socially oriented uses, based more on ability to pay, while "pay what you want" is often more broadly oriented to perceived value in combination with willingness and ability to pay.

An example of this is at The Arcola Theatre's website

Arcola reserves a limited number of ‘Pay What You Can’ tickets for performances on Tuesday evenings. The average ticket spend is £5, but there are no restrictions either way; you simply pay what you can.

Common in English is "Name your price", or "Name your own price".