We are migrating a database from MySQL to MongoDB for performance reasons and considering what to use for IDs of the MongoDB documents. We are debating between using ObjectIDs,
Consider the amount of data you would store in each case.
A MongoDB ObjectID is 12 bytes in size, is packed for storage, and its parts are organized for performance (i.e. timestamp is stored first, which is a logical ordering criteria).
Conversely, a standard UUID is 36 bytes, contains dashes and is typically stored as a string. Further, even if you strip non-numeric characters and intend to store numerically, you must still content with its "indexy" portion (the part of a UUID v1 that is timestamp-based) is in the middle of the UUID, and doesn't lend itself well to sorting. There are studies done which allow for performant UUID storage, and I even wrote a Node.js library to assist in its management.
If you're intend on using a UUID, consider reorganizing it for optimal indexing and sorting; otherwise you'll likely hit a performance wall.