Pen Pen's proper name, which I think is cooler in a geeky sort of way, is Pen². Pen-Pen, that's Pen squared, was once Fed Exed to a hot springs resort by Kaji Ryouji, where he witnessed Shinji Ikari's thermal expansion.

He comes from a newly evolved breed of Tropical Hot Spring Penguins. He was part of an experiment, but when the time came for him to go, Misato Katsuragi took him home. Fish is one of Pen Pen's favourite foods, however, unlike Antarctic Penguins, his diet also consists of beer, curry and ramen. He loves taking showers, and sleeping in Misato's fridge.

By the end of Neon Genesis Evangelion Misato had to send Pen² away because it was getting too dangerous around where he was. What happened to him during The End of Evangelion is a mystery.

Hideaki Anno's thoughts on Pen²'s name is as follows:

Super straightforward naming, but I thought the reduplication sounds cute. His official name is the 2nd power of Pen. I was reluctant at first, but it appeared the show because we thought we need a mascot animal. The show is set in a hot-spring resort Hakone, and one associates a monkey to a hot-spring. But that is no fun, so we decided to make it a penguin, most unsuited animal in a hot-spring. I remember "Hot-spring penguin" is Yoshiyuki Sadamoto’s idea.