Commerce product with customer defined attributes
Absolutely.  What you need is the Commerce Customizable Products module.

That allows you to set up Line Item Types that are entirely fieldable!  You can add text fields, select boxes, radio buttons and any field you want to be displayed as the user is adding the product to their cart.

Once the module is enabled, do the following


Add a new Line Item Type at /admin/commerce/config/line-items
Add any fields you need to it (in your case, just a text field)





Make sure to check the box that asks whether this is a field you want added to the 'Add to Cart' form





Finally, go to the 'Manage display' tab of the content type you have that displays your product, and go into the settings of the 'Product' field.  You will see a new option which allows you to choose a 'Line item type' to display.  Select the one you just created, Update, and Save!




Final result:



Let us know if this works for you...