Upvote:2
Your best bet would probably be to issue a formal complaint to the airline for not following correct procedure, then follow it up intensively. Keep hounding them as they'll probably try to bury it. That said unless you're a native this is probably going to be stressful and expensive (though hounding them by email and on social media is free).
If the lighter was valuable (a rare collectible, for example), you may have a decent argument (if perhaps not a legal one) for compensation of the value of the item or a replacement of the same model and vintage. Unfortunately, if the value was only sentimental then you're unlikely to receive anything more than a token gesture if anything at all.
As for your rights, that would probably depend on Mongolian regulations, and you'd probably need a native to find that out.