Product Attributes shown in the Add/Edit Product window and Product Multi-Select window will only be those explicitly marked to display at Hub / Setup / Company Preferences on the Product tab.
Select Product Attributes then add a new attribute or edit an existing.
On the Edit An Attribute window, select Show In List for that attribute to display in the windows mentioned above and select Save.
Once the attribute is set up appropriately, right-click a heading at the Add/Edit Product window and/or Product Multi-Select window and select Choose Columns (or Choose Columns Alphabetically) to check the attribute and have it display in the grid. From there, optionally right-click again and choose Make This Layout My Default.