I was wondering if there is a tool I can quickly install on a sharepoint production server to allow me to browse lists in a site, and display their names + id's?
Maybe overkill for you what you need, but if you haven't got it already, get SharePoint Manager. Allows you to view every object in a SharePoint Farm and view all it's properties.
Sharepoint Manager (as suggested by James) could be your first choice. If that is not allowed in production, PowerShell can help you.
Following command would give you what you need
$web = Get-SPWeb [URl of your web site]
$web.Lists | Select Title, ID