It happened to me once in London Heathrow. In my case I didn't have any emotional attachment to the item and it only cost around £3 to replace, so I threw it away, however I discussed with the security and the airline what my options were. They offered me the following options:
- Throw it away (I actually ended up doing just that)
- Post it to whatever address I wanted it - in Heathrow airport, there is a post office, although not in every terminal.
- I could put the item in a separate bag/whatever and check it in. If I didn't have a bag to put it into, there are plenty of shops landside that sell luggage (various bags, suitcases, etc.) - I could purchase the right size bag there, put the item in it and go back to the airline counter to check it in. Depending on the airline and how many bags with how much weight you have already checked in, they may charge you for this additional one.
- Refuse to fly - my checked in luggage would be returned to me, but I would forfeit the ticket.
The main point to keep in mind is that you're still landside. Depending on how much time you have left before your flight, you could even go back home, leave the item there and come back to the airport. For example, if you're in Heathrow and live somewhere near Paddington, you can easily get from the terminal to home and back to the terminal in under an hour.
That said, your options may be very much limited by what airport you are in.
If everything else fails and the item is REALLY, REALLY priceless, you can always turn around, go to the airline desk and tell them that you changed your mind and don't want to fly. They will then return you your checked in luggage. You pack your item there, then buy another ticket. Prepare to be quizzed by some men in black though :) "Changing your mind" after having checked in is considered a very suspicious move.
P.S. The item that happened to be in my backpack was an 8mm drill bit for glass and tiles, which I lent to a colleague, then he returned it, I put it in a small outside pocket on the backpack - and completely forgot about it. I flew with this backpack almost 3 months later - and ended up throwing the bit away, as buying a new one was the cheapest option.