Using words like "star" or "favorite" usually imply some sort of bookmarking or tagging, which in this use case would not communicate to the user the true meaning of the interaction.
The three examples below are similar in that they explicitly label the action using words that are unambiguous. Any variation on the wording below would be acceptable so long as a user who has never used your UI would be able to look at the element and tell you what it does.
To make a file or folder available offline Open My Computer.
My Computer, click the shared network file or folder that you want to make available offline.
File menu, click
Make Available Offline.
Follow the steps in the Offline Files Wizard.
Similarly, in Microsoft Office, the language explicitly describes the function:
Google Docs on iOS uses a slightly different approach:
To make a file available offline in the Drive, Docs, and Sheets apps,
touch the grey i button next to the file, within a file or the file
list, and switch the setting from Off to On next to
Keep on device
in the details panel.