You can save the document in the chatter of your lot/serial number. Doing so, when you will have to find back the document, you will find it by opening the lot/serial number. In fact, you can't "search" for a document that is saved in a chatter, so you will have to search for the lot/serial number itself.

Otherwise, you can try to use the Documents app. Doing so, you will be able to create a workspace named "Serial Number" and use tags to quickly find each serial number.