The caller is going to be typing in their credit card number into the IVR. However, we need that number hidden in the logs. I know that setting a variable to private can make it hidden, but what about the menu:value? I'm not able to go into project variables and make the menu:value to private since that variable doesn't exist in project variables. Would appreciate any suggestions.

You should be able to see the menu variable and value field in the project.variables editor, and set the Private property to "true". If not, there is problem with the project.