This is pretty straight-forward in InDesign.
Create a page with just the standardized text for each of your products and save it as a .indd file. For the actual product labels create a new document, and place the page with the static text (File > Place...) and it will create a link to the original. Add new text fields to hold the customer information, and save as an InDesign Template file that can be opened and customized for each customer.
When you need to update the static text, you edit the linked file, then update the link in the template, and in any customer files as you reopen them.
This is really rare - I'm going to disagree with Peter.
If you only have fifty customers total, then doing your content management in InDesign will totally work, along the lines that Peter suggested. But what happens if your business has a boom year and all of a sudden you have two hundred customers? Or five hundred? Of if instead of one variable (customer-specific product name) you suddenly have lots of variables (separate warnings for separate customers in different countries, permutations of product combinations, et cetera).
InDesign is a splendid page layout app, with lots of tools that will let you manage your content. But at some point - that you can't know about unless you're actually involved in the company - it can spin out of control due to the fact that the designer has to know about all of the content-management features that are being used by your specific workflow. (And then the designer will go on maternity leave, or whatever.)
So I usually point people towards InCatalog and InData as plugins that work really well with InDesign, that let them keep their rapidly fluctuating content - their business data - in a database, where it belongs. Database-driven workflows in InDesign can be easy or difficult, depending on the kind of data that's being instantiated in ID, and the tools used to get it from the db to the page. You have lots of options here, and it's pretty much impossible to tell you which one to pick. It could be simple (save .csv out of the db, use InDesign Data Merge) or hard (export XML from the DB, use InDesign's XML Import) or it could be a third party tool that works better for your given implementation.
All that being said - if the only variable in all of your text really is the name of the product alone, I'd point you at the native InDesign feature of custom text variables. You define the name of the product in one place, and all subsequent instances of the variable follow along. No plugins necessary.