Discounts are added to purchases and subscription products on a customer, rather than to the customer itself. The following are steps to add a specific custom discount to that customer's subscription products.
To add a discount to a subscription product:
Navigate to the customer's active subscription product:
Select the "Discount" option for the Advanced Overrides field on the subscription product and click "Go"
A pop up will appear, click "Add Discount"
Enter the values for the fields, click "Add" and click "Apply"
Note: Periods is the same as the frequency of the subscription. E.g If it is a monthly subscription then the period is in months
Save the changes for the subscription product
Information on how to add discounts to multiple customers at a time can be found here Creating and Applying Discounts and Coupons Overview