Here's a quick run down of how to enable end users to edit distribution lists, this saves you time and gives department managers greater flexibility.
Go to your company or clients Exchange Admin Centre, go to Permissions then user roles and create a RBAC by hitting the +, give it a good name and description. Under the Distribution groups section, tick the "MyDistributionGroups" option and any others you might need and save the RBAC, the default RBAC has all but the MyApps selected, remember this will replace the Default Role Assignment Policy for the user you assign your RBAC to.
Now under recipients in the Exchange Admin Centre find the user and edit them, under mailbox features change their Role Assignment policy to that which you've just created and save.
Now whilst still under recipients, click on Groups, find the group the user needs to manage and edit it, under ownership add the user to is to manage the group and save.
Now the user should be able to open Outlook, click Address book, find the group and add/remove other users.