Using functions.php inside your theme will have the functionality only for that theme.
I think a simple approach will be to create a plugin or a mu-plugin as @Benoti is suggesting where you will create a settings page that will be saved on options table.
There are a few guides to create admin pages:
https://codex.wordpress.org/Creating_Options_Pages
There are also some ready made plugins like this ones: https://wordpress.org/plugins/search.php?q=options+page
From all available options I think is best to create your own options page as it is not very complicated and make it as a mu-plugin so all sites in your network will have it without the need of activation.