The Arkham Horror errata says on page 6:
Q: Some Items state that they cannot be lost or stolen unless you choose to allow it. When does this ability come into play?
A: It would come into play when an investigator goes insane, gets knocked unconscious, or defaults on a bank loan. It would also come into play when any other game effect specifically instructs you to lose an item or tells you an item is stolen.
This answer is vague. Since you are choosing which items to lose when you go insane or unconscious, the only way I can see this answer making sense is that they are claiming these particular items could cause you to not lose half of your items (rounded down).
My interpretation is that if you have two items, both of which can't be lost unless you allow it, then you can retain both when you go insane/unconscious. What if you have 4 items that can't be lost and 2 that can. Do you get to lose only 2 of the items in this case? If you have merely 1 item that can and 1 that can't, is it basically no different than if you had regular items (and would mean you are simply choosing the one to get rid of)? This leads me to think that this quality of items only really comes into play if greater than "half of your items rounded up" have this quality. i.e. 3 out of 4, 4 out of 5 items, 4 out of 6 items, 5 out of 7 items, etc, in which case you could avoid losing the full "half rounded down" amount. Do I have this right?