But let's take a step back and review what we DO know about her.
In Season 3, we first encountered Ms. Hawking interacting with Desmond in London during one of his flashes back in time; she worked at the store where he went to purchase an engagement ring for Penny ("Flashes Before Your Eyes").
Desmond choose a ring, but for some reason Ms. Hawking was adamant that he was not supposed to take it; that it was not his fate, because doing so would alter the course of the world (every single person will die if he does not crash on the island and press the button every 108 minutes).
Desmond was understandably bewildered, so Hawking took him outside and told him to look at a man wearing red shoes (yet another Lost literary nod to The Wizard of Oz). That man was suddenly killed in a construction accident; she wanted to prove to Desmond that it was that his fate to die and that the universe has a way of course correcting itself.
I also believe it is significant that Ms. Hawking was wearing an Ouroboros pin during her encounter with Desmond. I first became aware of this symbol when Scully got a tattoo of one on The X-Files (nerd alert). There are many interpretations of what it represents, but most versions of this symbol feature a snake swallowing its tail, illustrating return (to the island?) and the cyclical nature of life. However, Hawking's serpent is not doing so. The open nature of her Ouroboros may indicate her ability to travel back and forth in time without getting stuck.
Later on in Season 3 ("Catch-22"), we catch a quick glance of Ms. Hawking in a photograph with Brother Campbell at the Monastery where Desmond briefly lived.
At the end of episode 5.02 ("The Lie"), we witness Ms. Hawking working out equations on a chalkboard and archaic computer while a large pendulum maps out corresponding coordinates and determines an 'event window.'
It appears as if her workspace is beneath a church, and when she emerges upstairs from the underground physics lair, we are surprised to see Ben awaiting her arrival. Hawking seems to rank higher than Ben (at least off island), and is responsible for ascertaining the 70 hour time line for the O6's return to the island.
And now, I will completely contradict many of my previously posted statements and theories.
Fact? Eloise Hawking is Daniel Faraday's mother because...
- She lived or worked at Oxford, where Daniel was a professor
- She is proficient in Physics, his specific area of expertise
- Daniel's lab rat was named Eloise
- She appears to have the ability to travel through time, like Daniel
- He may not have outright recognized her as his mother, but Daniel did say that she resembled someone he knew
- Did young Ellie hook up with young Widmore? Are Charles Widmore and Ms. Hawking father and mother to both Faraday and Penny?
- If Faraday is Hawking's son, why doesn't he have a British accent? He was a professor at Oxford but we don't know where he grew up (only that he 'fled' to the States after something happened to Theresa Spencer).
- When did she leave the island, and why? If she and Widmore were expecting, is that why both of them left the island (so the baby would live)?
- Could she and Widmore actually be siblings instead?
- Has she been back on the island since that time?
- Is Faraday ageless, like Alpert? If Ellie is his mother and she had him around 1954, he would have been 50 years old (give or take a few years, depending on when she had him) when he landed on the island from the freighter.
- On the island, Ellie and Widmore were hostiles/others under Richard Alpert's leadership...at what point does Widmore switch to the dark side (opposite Ben, and thus Alpert), and how does that affect his relationship with Ellie?
- Is Hawking working with Ben because she wants to get Daniel off of that island?
- Does Penny know whether or not she has any siblings? Does Penny know who her mother is?
- Desmond and Penny are off to Los Angeles in search of Faraday's mother. If that turns out to be Hawking, they'll more than likely also encounter Ben. Given that Ben told Widmore he was going to kill Penny...how will that play out?
- Does Hawking know that Widmore 'changed the rules' by having Ben's daughter killed, and that Ben now wants to kill Widmore's daughter?
- Does Widmore keep in touch with or still see Hawking? How does he know that she's in Los Angeles and know where she lives?
- Is she able to predict or see anyone else's future besides Desmond? Does she know all about the pasts and futures of the O6? Did she visit anyone else in the past, to put them on the right path toward their destiny (like Abaddon urging Locke to go on the walkabout)?
- Did Hawking arrange (via Brother Campbell) for Desmond to live/work at the monastery, to meet Penny there? Was she aware of his fate/destiny/role early on? Did she know that Desmond would be Faraday's constant?
- If Faraday is her son, was he feeding her information about the island after his time travel there and back? Were they working together yet apart in time shifts to course correct the universe? Was he telling Widmore as well, given that he was funding his research?
1. Hawking is an island emissary. She is not on either Ben's or Widmore's "side," but represents the best interests of the island. As a young woman, she was on the island working with Alpert and Widmore to save the island from the detonation of the hydrogen bomb. As an adult, she is working with Ben to make sure that the O6 return to the island in order to stop the 'very bad things' occurring there.---
2. Ellie is not Hawking, she is Danielle Rousseau. Same cantankerous countenance and penchant for pointing guns first and asking questions later.
There are endless explanations and theories about who Ms. Hawking is, who she is related to and how this will all unfold. I certainly don't claim to have the answers, but as I've said before - the fun is in the sheer speculation.