If you want to edit the color of a specific attribute in a block reference (insert), you can use the standard functions for editing block attributes.
The most convenient is the EATTEDIT command - this command is also invoked when you simply double-click on a block reference. Go to the Properties tab (the 3rd one) a select the new color for the given attribute in the Color list:
In the older versions of AutoCAD and AutoCAD LT you can also use the commandline version of the attribute editor: -ATTEDIT. Confirm the default Enter on the first 4 prompts, then select the attribute and use the Color option to set it to the new color. You can also use the menu macro:
If you want to change attributes for all references of the given block, edit its definition (BEDIT, REFEDIT).