The Club Nintendo replacement is called "N-Box," and it will be a subscription-type service. 
It is currently only available to those in the UK.  There has not yet been any information on releases for other countries.
The N-box will work like Loot Crate, which actually inspired the plan due to previous collaboration on an Amiibo box. The user gets a mystery box of various Nintendo-related items like clothing, toys, etc. once a month, depending on the chosen subscription plan. There will be 6, 9, and 12-month subscription plans. Each box is available on a one-off basis.