In SharePoint 2013 on-premise environment “Design Manager” option is missing as part of Site Setting gear icon.
In SharePoint 2013 “Designer Manager” will comes by default as part of publishing site collection, if the site collection is created with team site template you need to enable publishing features.
Follow the below steps to enable “Design Manager” in SharePoint 2013 on premise environment.
1. Click on site settings gear icon and navigate to site settings
2. Enable Site collection features as part of Site Collection Administration
3. Enable “SharePoint Server Publishing Infrastructure” feature
Once the publishing feature is enabled as part of Site collection features navigate back to site settings (step 1)
4. Click on “Mange Site Features” as part of Site Actions
5. Enable “SharePoint Server Publishing” feature
6. Now if you click on Site settings gear icon you can see “Design Manager”
Hope this helps, happy development in SharePoint!!!