NFTs are unique digital tokens which are often attributed to art that has no physical representation.
Chloe Clem, now 10, became famous in 2013 when her mother shared her unimpressed reaction to a surprise Disneyland trip online.
The image has become a popular meme for expressing concern.
An NFT provides a digital certificate of ownership for the piece, but bears no copyright. Therefore the winner of the auction will own the digital token representing the meme, but not the meme itself.
Within weeks, a Buzzfeed article had declared Chloe "the patron saint of Tumblr" and "the queen and goddess of the internet".
Now the Utah family has decided to auction off the image as an NFT. Bids start at 5 Ethereum - a form of cryptocurrency worth roughly $15,000 (£11,000).
Ms Clem said that after learning about the NFT market, selling the meme was a "no brainer".
"It's a cool opportunity, especially if there's a Chloe fan out there who loves this meme they'll be able to own it. Even Chloe has said 'that's pretty cool' - it's a typical 10-year-old thing to say."
The market for ownership rights to digital art has boomed recently after several multi-million-dollar NFT sales.
In March, Twitter founder Jack Dorsey sold his first ever tweet for the equivalent of $2.9m to a Malaysia-based businessman.
And in April, the woman featured in the so-called Disaster Girl meme sold it for $500,000. Weeks later, another image called the Overly Attached Girlfriend meme sold for $411,000.
In August, however, a fan of British graffiti artist Banksy was tricked into buying a fake NFT by a hacker, off the artist's website. Luckily for him, the hacker returned most of what he paid - more than $336,000.
