I'm attempting to add more languages into a custom Wikibase installation as shown here: 
In the area where "Language" is I should be able to list other languages. I've clicked "Configure" which links out to the general MediaWiki help for configuring languages (https://www.mediawiki.org/wiki/Help:Extension:Wikibase/Configuring_languages). However, it doesn't seem to explain how to do it on a custom installation.
I did a bit of digging online and discovered that maybe installing the Universal Language Selector extension was the way to do it, but I've installed it (and verified the install) and it still doesn't seem to be possible.
Any help on this would be much appreciated!
I'm not fully sure if I understand your question. Are you maybe looking for https://www.wikidata.org/wiki/Special:SetLabel ? It lets you set the Label for any Item or Property in any supported language.