How to add a new tab to Case Activity tabs in Salesforce?

Whats' the custom tab? Keep in mind those tabs are meant for specific actions. Here's the trailhead that should help:

Actions and Lightning Experience:
https://developer.salesforce.com/trailhead/lex_migration_customization/lex_migration_customization_actions

Here's a quick summary of how you'd customize the actions:

You'll be creating your quick actions under the Buttons, Links, Actions options for the respective object (New Action). In your case, you would go the the Lead object under settings/object manager, then create a new Action, which will be whatever the custom action you'd like to do.

Once created, you can go to the layout of the lead page and add that Quick Action to the area in your layout called "Salesforce1 and Lightning Experience Actions." You'll want to add as one of the first 4 actions for it to show in your layout.