Can I print different brand names for the same item in BUSY? If yes, how?

You can use parameter feature to specify brand name and to show on print but you need to follow some steps:
Step 1: Go to the Administration and then click on configuration.
Step 2: Now click on Feature and options and then click on inventory tab.
Step 3: Now enable Parameterized Details option and can also search it from F3 short key.
Step 4: Now click on Configure option and specify the number of parameters as 1 or if you required more so you can specify and click on configure option.
Step 5: Now specify the Name as Brand name in parameter 1 and if you select open then you need to enter brand name during voucher entery and if you select Predefined then you have add values there for brand and can pick it during voucher entry from drop down.
Step 6: Now click on ok option and also can enable option as per your requirement in paramaterizes stock details options window and again click on OK option.
Step 7: Now save the inventory tab and then Go to the Administration.
Step 8: Now click on master and then select item option.
Step 9: Now click on Modify option and select the item of which you want enter brand.
Step 10: Now enable Paramaterized details in item master and save the master.
Step 11: Now Go to the Administration and then click on configuration.
Step 12: Now click on invoice document printing option and select standard option.
Step 13: Now select the Invoice type and enable parameterized option to print.
Step 14: Now go to the transaction and enter purchase or stock journal and enter brandname after item and quantity in voucher and save it and then print from standard format.