The Last of Us Part 2: Remastered's Jackson Dance lost level gives a bit more detail about how Ellie got her signature moth and fern tattoo, including showing the character who gave her it.
One of the most notable bits of iconography from The Last of Us is Ellie's moth and fern tattoo that she gets in between the events of the first and second game to cover up her bite mark. As iconic as the tattoo is, it's not focused on all that much in the original release of The Last of Us Part 2, although it originally had a bit more backstory to it.
One of the biggest things that The Last of Us Part 2 Remastered introduces is the Lost Levels, a set of unfinished levels that were cut from the main game. This includes the Jackson Dance level, which would have expanded on the original game's flashback scene that showed Ellie and Dina kissing for the first time.
Beyond adding some more depth to the Jackson segment of the game, the Jackson Dance level would have also given us a bit more background on Ellie's tattoo, including introducing us to the character that designed it for her. Early on in the level, Ellie could have stopped to get her face painted by someone called Kat, who is an ex of Ellie's and is revealed to be the person who gave Ellie her tattoo.
The short sequence between the two in the Lost Level isn't voiced, but we do get an idea of how their interaction would have gone. If Ellie talked to Cat, she would have asked her how her tattoo is doing, before suggesting that she touch it up and add more to the design, something that Ellie doesn't seem that keen on.
Banun Idris, a game designer at Naughty Dog, offered some developer commentary on the scene with Cat, noting that, "Halley [Gross], our writer, mentioned that in her mind, an ex-girlfriend of Ellie's tattooed her arm to cover up her scar. So we figured that this might be the only time you get to see that. We wanted to hint at it and allude to it and make it feel like there had been so much more happening, and so much more that Ellie threw away."
This is something that is actually hinted at in the original version of The Last of Us Part 2. If you look at Ellie's journal during flashbacks, you can see Kat being mentioned as Ellie's girlfriend at several points, as well as confirmation that she's the one who designed her tattoo. While Kat didn't make it to the final game like the Jackson Dance would have allowed, at least her impact on the story remains in the full release.