"Pirate Asuka" was first seen in the previews after the second film, and finally featured in the third, but how did she end up losing an eye?
The second Rebuild of Evangelion movie ended with a tease for the next film, but that movie wouldn't be out for five years, leaving fans with a mysterious image of Asuka wearing an eyepatch that led to a lot of jokes. While Asuka does have an eyepatch in the third film, it's kind of glossed over, leaving many fans of the original series to wonder how exactly she got it.
Among the many changes for Rebuild of Evangelion, one of them was eliminating Toji Suzuhara, Shinji's friend from school, as a potential EVA pilot. However, the events related to Unit 03 and the Angel still occur--this time, however, it's Asuka in Unit 03's pilot seat. Unit 03 is contaminated by a blue goo, which turns out to be the Ninth Angel. The goo reaches the inside of the entry plug and begins to contaminate Asuka as well, causing an emergency alert as the Eva activates and begins rampaging. Unit 03 is now indistinguishable from an angel, and the former entry plug is now recognized as the Angel's core. Unit 01 ends up destroying Unit 03, ripping the Eva to shreds, but Asuka (somehow) survives.
Asuka Has an Eye Patch Because Unit 01 Attacked Her
When Asuka is next seen in Evangelion 3.0, she is wearing an eyepatch over her left eye, although it's not addressed. The initial assumption one would make is that it's an injury from the Unit 03 incident, which is at least partially true. However, later in the film, when Shinji and Kaworu are descending into Terminal Dogma in Unit 13, Asuka tries to stop them in her upgraded Unit 02. Failing to do so, Asuka unleashes her last-ditch effort: she takes off the eyepatch and reveals a "sealing pillar" has been implanted where her eye was, and that the Ninth Angel was never truly purged from her body. Removing the seal, the Ninth Angel takes hold and Asuka (and Unit 02) transforms. Unfortunately for Asuka, this response from her was part of the plan, and her Angel core (the entry plug) is absorbed into Unit 13.
From a meta-perspective, there's another reason why Asuka has that eyepatch. In End of Evangelion, Asuka and Unit 02 are defeated by the Mass Production Evas, which are equipped with duplicates of the Lance of Longinus. One of them throws the Lance at high speed, which pierces Unit 02 (and thus the synchronized Asuka) through the eye. The eyepatch in Rebuild of Evangelion also doubles as a nod to this iconic scene, and perhaps even acts to convince dedicated fans that they know what happened to her eye, only to be surprised by the Ninth Angel reveal. It also helps to show that a fair amount of time has passed, despite Asuka and the other Eva pilots not having aged.
"Pirate Asuka" certainly was the focus of some memes around the time of the eyepatch's reveal, but Neon Genesis Evangelion has taken what first seemed a bit silly and made it properly horrifying, just as the series is so good at doing.