If you are looking to offer a discount on a customer's existing membership, the best way is to provide it on a one-off basis. Discounts applied to memberships affect all contract holders rather than just the one they wanted to discount.
On a one-off basis, the best option is to apply a complimentary account balance to their account and ensure "Use Account Balance" is turned on inside their profile.
Another option would be to provide a Line-item discount. Line-item discounts will only apply to the first payment interval of the membership. Also, the membership must be set to Bill-On Purchase.