You would need to give the permission "System > Administer site configuration". You can give that permission basically to whatever role you like. This lets users with that role put your site into Maintenance Mode. But be aware that this is a **very powerful permission** and it comes along with other administrative rights. Also – if you give that permission to a role you should probably also give the permission to view the site, if it's in Maintenance Mode. 

I would consider such practice actually **a security risk** and would recommend to only give that permission to an extra role like "co-admin" or the like.

Edit: there's a module that let's you split administrative permissions into smaller chunks called [Custom Permissions][1]. The project page also gives a good impression which administrative rights are by default connected to the permision  "System > Administer site configuration".


  [1]: https://www.drupal.org/project/config_perms