FWIW, the 409A prohibition on extensions for NSOs and disqualifying amendment rules for ISOs would seem to me to make this a nonstarter. Seems to me the option has expired / terminated based on the clear terms of the grant notice as described above. From that description, it sounds like there was good reason / clear intent for including the earlier expiration date. I'm guessing the date just snuck up on the optionee?
If that is not the case and there is some support or basis for thinking the original terms were intended to have the option expire 12/31 in all cases instead of the earlier date and you can show some clear support (e.g., board resolutions / minutes, etc.), then maybe you have an argument that this was a scriveners' error / typo and the option is still exercisable? That seems a difficult and thus aggressive argument to make though.
Was this an ISO such that there would be no taxes triggered on the exercise? If so, hard to replicate that treatment but if this was an NSO (and/or the company wanted to try to provide some similar benefit to the optionee) then maybe you provide them with a restricted stock award now (fully vested) with the right to purchase the same number of shares for a purchase price equal to the option exercise price through December 31st. That should put the optionee in the same general position as exercising an NSO. Company could grant the optionee more shares and/or lower the purchase price and/or provide a bonus, etc. to the optionee to help offset immediate taxes if it was an ISO and company feels responsible or sympathetic, etc.. (Without knowing the facts though seems the onus should be on the optionee to know the terms of the award and its expiration date.)