are you using NX in Native mode or with Teamcenter?
You could try the following: menu->assemblies->cloning and under Naming Tab, select the Define Naming Rule and check if "Add Suffix" is selected. I think it's called Numbering insead of Naming if used with Teamcenter.
Also check the customer defaults, go to Teamcenter Integration for NX --> Part File Name Display and check if a suffix is added.
Try @MATHI_CAX_TRG's suggestion, but in the text to replace option enter the common suffix "_X" (without quotes) and in the replacement text, leave it blank. If it won't allow a blank entry, try entering a space character. A space character at the end of the file name is usually trimmed off before adding the file extension.
Ok, try this:
Text to replace is "_X". In the replacement text entry, enter a single character such as "~". Specify an output log file and do a dry run (don't do the actual clone). Close the clone dialog. Find and edit the log file (use a plain text editor such as notepad). Near the top of the file, you'll find the naming rule:
&LOG Naming_Rule_Type: REPLACE_STRING Base_String: _X Replacement_String: "~"
Delete the replacement string to create an empty string:
&LOG Naming_Rule_Type: REPLACE_STRING Base_String: _X Replacement_String: ""
Save the log file. Now open the clone dialog and use the "load and apply existing log file".
Edit: I tested this on NX 9. Should also work on NX 11?