If I create a new object with a lookup field in lightning experience, I can search for / pick the object based on its name field:
However, I'd like to be able to search on other fields from directly within this dialog, and have another field displayed here other than the name field. Is this at all possible?
You'll need to set-up the Search Layout Configuration options for the New Object in Lightning Setup, including the Lookup Dialogs and Search Filter Fields. See Configure Lookup Search Results in Lightning Experience for more details.