Answer
To make a custom field itself searchable, open the Custom Fields window, right-click the relevant custom field and choose Edit to open its configuration wizard, and enable searchability there — once synchronized, the field can then be added to that LU's own search dialog. Surfacing a custom field as an option specifically within an existing standard field's LOV popup is less certain and not fully confirmed as generally supported, but two things worth trying: include the LOV's own underlying view in that LU's Approved Detail Views list (you can find the exact LOV view name — for example, INVENTORY_PART_LOV_MRP for the Part No LOV — by keeping the Debug Console open while clicking the LOV button), and separately check the dedicated 'LOV' checkbox available on each custom field's own configuration, which is a simpler, more direct toggle than opening the full wizard.