To add a gift registry, we will need to install a 3rd party shopify app. The Gift Registry app that gets the most positive reviews, and from my research appears the most full featured main go-to app is called Gift Reggie.
This is a premium app, and depending which plan is used, ranges from $5-40/mo.
Only pay for the registries you need. All of our plans have no limit for the amount of archived/closed registries you have, and we offer flexible pricing based on how many registries you'd like to be available to your customers at a given time.
For more details, please visit gift-reggie.eshopadmin.com.