I've spent days trying to find out how to save or update a value into a CustomField programmatically and finally found out how it's done. So I'll make this a question and then answer it as I would have loved to have this question and answer.
There is conflicting documentation on how to save or update a value for a Custom Field in JIRA. I was using:
customField.setCustomFieldValue(CustomField, value);
This does not save the value into the database but it does update the value as far as I can tell. It's only useful if you are using the CustomField further down in a Workflow Post Function transition for example.
I'm using Jira 4.3.2.
How do I persist the the CustomFields value into the JIRA database?