Summary:
The custom field created in the MS Projects Custom field section, even if enabled in the task(s) of the project,  may not show up in the MS Project integration managers Field mapping section.

Cause:
Since the MS Project integration manager is mapped to folder containing the MS Project files, it reads all the files in the folder. If any of the project files in the folder does not have the custom field enabled in it, then the integration manager would not display the custom field in the drop down. The integration manager expects the custom field to be enabled for all the projects in the folder.
 
Resolution:
1. Have the custom field enabled for all the project files available in the folder mapped to the MS Project integration manager.
2. If you would not like to have the same custom field enabled for all the project files, then move rest of the project files to another folder, so that the integration manager detects only the specific project file with the custom field.