How to pass a Green Card through immigration to someone arriving to the US from abroad?

My wife has arrived now and passed immigration successfully, so I can answer the question based on this one experience, although it won't necessarily be the same for everyone.

The direct answer - the green card wasn't needed in the end. She had a scanned version of the green card - the immigration officer said they don't accept scanned versions and didn't even look at it.

I called the airport and spoke with an immigration office before arrival. He was quite friendly, and said if she forgot the green card, they do have other options. He did emphasize, however, that he can't guarantee her entry until he meets her and has a discussion.

When my wife arrived, she spoke with this same officer. She said it didn't take very long (about 5 minutes), but the whole process of waiting in immigration - being sent to a private room to meet with this officer, and getting his approval - took long enough that she missed the connecting flight. She had a 1 hour 20 minute layover.

It seems to me that the website wording is intended to appear strict to avoid such situations, but in reality flexibility exists. Calling before hand seemed to help show that it was an honest mistake.

It also seems that the result is up to the officer's discretion; luckily our officer seemed to be having a good day.

As some comments mention - if this happens to you, the hardest part will be boarding in the first place. We were fortunate in that her passport had a visa that was dated 2 years in the future instead of 1 year as it should be.

In addition to the advice given in Hanky Panky's answer, if she has a smartphone or other portable electronic device that can access a data network with her, it can't hurt for you to take a picture of her green card and send it to her. They'll probably still want to see the real thing (and, thus, Hanky Panky's advice of trying to meet at the airport with the actual green card is still good advice,) having a picture of the card in her possession can only help her case.

I cross US-Canada border 3-4 times a week for my work. Once I forgot my Canadian PR card and only realized after crossing into the US. On my way back, I explained the situation to border officer. He told me to park and speak with a different officer. I showed them my driving license and passport and after few simple questions, they allowed me to enter into Canada after verifying it from their systems.

What I was told by border officers which also applies to US green card is that these cards are the important proof that you are a legal resident, BUT they aren't the only proof. If you can prove to officers that you are a legal resident by some documents (like driving license), they will have to let you get into the country because that's where your home is, where else would they send you? But these are extraordinary situations and should not be misused.

Edit: This answer made sense for the original question but is not relevant after that was edited.

Without evidence that she has the right to enter the USA, she may not be allowed to board her flight in the first place.

I would use a courier such as FedEx and send the card to her.

EDIT: apparently the temporary I-551 was expired, in which case beats me how she got on the plane

Old answer:

The CR1 visa, even if expired, can be used for entry instead of a green card for a year after the first entry on it. Check-in staff knew this (it even says on the visa) and thus let her board.

So she'll be just fine, and you don't have to do anything.