Electra was originally invented as a joke. When Edrobot was applying to a Nasuverse-themed RPG, he deliberately suggested what was, in his own words, "the worst possible idea for a Servant"; a hypothetical future daughter of the characters Saber and Gilgamesh , from Fate/Stay Night. Though this concept was rejected in favor of a much more reasonable Servant, Edrobot recycled Electra's design and personality as a supporting character for an RPG he himself was running (though her backstory was changed significantly). She later became a supporting character in Jake English's Mysterious Theater of Scientific Romance from the Year 3000, and finally, a villian in Justice Society Of Japan.
Despite sharing a name with a mythological figure, Electra Pendragon has nothing to do with the greek Electra, nor the Marvel character of the same name. Edrobot has said that he picked her first name because it sounded cool, in keeping with the idea that she's the worst Servant concept ever.
As one would expect from someone raised by Gilgamesh, Electra is stubborn, haughty, and considers herself high above the "plebeians" that surround her. She's also very naive, and is prone to flying into a rage when confronted with failure, and is generally opposed to teamwork, preferring to do things by herself. As a result, her abrasive personality repells pretty much everyone she meets. This holds true for both versions of the character; the difference is that while her sociopathy is played for laughs in JEMToSRtY3k, in Justice Society of Japan her flaws are played for tragedy.
Powers and Abilities
Electra is stronger and more dexterous than a normal human, but is uncharacteristically weak for a Servant. She only appears to have a single, unnamed Noble Phantasm, which lets her enchant any object she touches, changing it's appearance and making it far more powerful than normal. However, when she lets go the item falls apart almost immediately. Worse yet, she seems to have poor control over her power, and has once accidentally enchanted the bonds she was struggling to free herself from.
She also seems to have a knack for engineering. She once claimed that a device called an "apoko-scope" was something she invented, and later on builds a makeshift grappling hook as part of an attempt to escape from a mineshaft.
Electra's servant class (if she has one) is unknown.
Role in JEMToSRFtY3k
Electra Pendragon is intially introduced at the beginning of season 2, alongside Alfred Smith and Lant Kreck, two other OCs invented by Edrobot. They had intended to use a time machine in place of Karkat, Jake, Dave and Pinkie Pie, who had all been seemingly killed by Caius Ballad at the end of season 1. However the four heroes turned out to be fine, and took Alfred's time machine for themselves. During Season 2, Electra mostly hangs out with Al and Lant, calling them her minions and trying to boss them around. She gets a more prominent role in Season 3, where the villainous Prohpet Gilesbie attempts to woo Electra onto her side, but she refuses, as her Father would never forgive her for serving under Gilesbie.
In season 4, she suddenly remembers that she was sent back in time to stop a catacysm known as "The Great Disaster". In a direct parody of the DC event comic "Countdown", she, Al and Lant (as well as a Gardevior named Opal) set off with Monitor Bob to find the internet reviewer known as Phelous because he was the key to stopping the Great Disaster for some reason. But when Bob turns traitor, revealing he was working for Monitor Solomon the entire time, Electra comes up with the idea of tricking Alfred's evil alternate future self into killing both of the Monitors, and then using a seemingly sarcastic suggestion Dave Strider made to kill said evil future version of Alfred.
Initally, it was implied that Alfred only tolrated Electra because he was sexually attractive to her. But during Season 4, Al revealed that his real intentions are to give Electra a friend, so that she doesn't end up like her alleged father, Gilgamesh. Though Electra herself never learned this, she had been gradually mellowing out over Seasons 3 and 4.
Gilgamesh himself made an apperance back in season 1, but despite threatening to destroy the world if he was ever summoned to the present, he has yet to appear since.
Role in Justice Society of Japan
Electra Pendragon is an antagonistic P.O.V. character in Justice Society of Japan. As in JEMToSRFtY3k, she was sent back in time to prevent the end of the world. She initially joins the titular team during a recruitment drive, but turns on the team when her "Apoko-Scope" predicts that the end of the world will be caused by Shirley Fenette (from Code Geass) as well as an OC named Ino.
Thanks to the efforts of Sayaka Miki, she was trapped behind a Witch's Barrier with no means of escape, though she was later captured by mysterious villains working for a being known as "The Witch Queen".