Tira here, from Shopify!
That's a very interesting offer to provide! Are these customers who were loyal to your store, or first-time customers, or something else?
I recommend setting up a discount code to provide a 100% discount off of the product that you want to provide for free. You can then give this discount code to the customers that you want to use the code. You can even make the discount code apply to only specific products/collections & specific customers, which allows the customers to login to their account, and then they can pop in the discount code at checkout. To find out how to create a discount code for a fixed value or percentage, check out our help doc: Creating discounts.
This way, you can provide your customers with a product that doesn't cost them anything. The one limitation of this suggestion is that if you don't want the customer to pay for shipping either, you will need to make shipping free for that product going forward. To set-up free shipping, you can go into Settings > Shipping and Add a Shipping Zone and add a weight-based or price-based rate, and select the option for free shipping. You can find all the instructions in our help doc on setting up shipping here: Set-up free shipping.
Let me know what you think of this, and if there's something else you're looking to do with this idea!