You can't remove the field, nor can you really do anything else to it. However, if Product Name is unimportant, I'd suggest the following setup.
First, go to Setup > Customize > Tab Names and Labels > Rename Tabs and Labels, edit Product, and on the second screen, overwrite the Product Name field with the values Product Code and Product Codes, for the singular and plural labels, respectively. I'd consider renaming Product Code to something else, like Product Code*, so that some screens that don't show API names are not as confusing.
Second, create a workflow rule/trigger/whatever that copies the Name field (which will appear as Product Code to users) to the Product Code field. This will help out in certain scenarios where the system is already using Product Code to do something.
Third, optionally, you can create a new Product Name field, or just skip that and use Description. It's your customization, you can do almost anything you like. It's okay to not want to use both a product code and a name, and there's nothing wrong with that, if that's how you operate.
Fourth, you might need to move data from the Product Code field to the Name field via the data loader so everything appears correctly. Once done, you'll probably want to remove Product Code* from your page layouts, etc so the data isn't redundant on the screen.
Finally-- and this is important-- document the changes so that future admins and developers won't get confused. We're not renaming API names, just the labels in the system. That's all fine and dandy for normal users, because they don't use API names, but admins working with formulas and so on, plus developers writing code, will need to know about this customization.
If this sounds like too much effort to maintain (and I can't blame you, I wouldn't want to), you can just train your users on how to live with this limitation. I know that it's not necessarily ideal, but you'll have to accept the lesser of two evils. Also, for what it's worth, you'll want to vote on this idea. It doesn't have any traction yet, but vote on it, and spread the word.
Also, this thread mentions that you can train your users to manually remove the name field from their search layouts, and it'll remain that way. You just can't remove it globally as an admin.