Answer
If a descriptive name field like this isn't offered in the search configuration (commonly because the visible value is actually resolved live from an ID via a lookup, rather than being a genuine stored/indexable field), create a custom field that resolves and displays that same descriptive name directly, using the appropriate API getter function to convert the underlying ID into its name. Once added as a genuine custom field on the screen, that new field becomes available to add to the search configuration the same way any other real field would be, making the descriptive name itself searchable even though the original built-in field wasn't offered as a searchable option.