Well first the alien shows up in Lain's room with a grey's head, but then later Lain shows up as the alien to Arisu/Alice. Just before erasing everyone's memory, if my own memory serves. This is significant.

Given the Gnostic context of SE:L, I'd say that the alien represents Higher Sophia.

I'll drop a quick rundown of some basic Gnostic cosmology here, because it's going to come in handy.

First you've got the Pleroma (the Fullness). This is basically the "real world". The Pleroma is made up of/inhabited by Aeons, which are sort of beings and sort of concepts. Collectively, they serve as a limited, manifest approximation of a deity that itself has no limits or borders. The Pleroma is created by what Kabbalists would call Ein Sof (and for which alternate terms are currently escaping me) through what's known as the power of limit. Basically by limiting aspects of its own omnipotence and omniscience, creating an approximation of itself that's split into these Aeons, which are distinct but connected to one another.

Now all of the Aeons in the Pleroma come in pairs. These pairs, together, not only sustain themselves and further generations of Aeons but they create the fabric of the Fullness itself. Each Aeon pair emanates to create more Aeon pairs in the same way that Ein Sof emanates to create the Pleroma. Sophia, one half of an Aeon pair, breaks off to go emanate by herself, though. This is a huge mistake.

Because Sophia emanates without her compliment she creates a realm known as the deficiency and a being known as Demiurge. Demiurge is corrupt, insane, and has no idea that the Pleroma or the Aeons exist. All it knows is the deficiency, a universe in which it has seemingly omnipotent power and which it believes it created, labeling itself God. Demiurge then creates the Earth within the deficiency and sets about being an angry and vengeful deity.

If you look at what Christians would call the Old Testament in Hebrew you'll see a few different words used to refer to God. Three of the most common are YHVH (Yod Heh Vav Heh, known as the tetragrammaton, this is where we get Yahweh and Jehovah), Elohim (which is, interestingly, both singular and plural as well as both feminine and masculine), and YHVH Elohim. These are usually translated as the Lord, God, and the Lord God. Gnostics generally contend that the things attributed to YHVH and possibly YHVH Elohim are actually Demiurge, whereas Elohim alone is the proper God who emanated the Pleroma.

Moving forward. So not only does Sophia create the deficiency with Demiurge ruling over it as its false god, she also splits herself in half and gets partially trapped within it. Now we've got Higher Sophia and Lower Sophia. Lower Sophia is stuck in the deficiency (on Earth) like the rest of us and subject to the result of Demiurge's insanity. Higher Sophia, though, is still in the Pleroma, and she's got a plan. She works together with either her Aeon partner, Christ, or a new Aeon pair that's emanated, Christ and the Holy Spirit. They're able to manipulate Demiurge into doing things to advance their cause and they also plant spiritual seeds within humans so that they can basically repair (literally re-pair) the souls on Earth and redeem the deficiency creating a whole bunch of new Aeon pairs and a smoothly operating universe.

This connection between the fallen soul in the deficiency and their Aeon partner in the Pleroma occurs through a sort of enlightenment experience known as Gnosis, thus Gnostic. Christ reconnects Lower Sophia to Higher Sophia and together they set about the work of redeeming humanity through Gnosis.

So, after all that, who is the alien? Who is Higher Sophia? Lain of the Wire.

Lain herself is Lower Sophia. She's an essentially omnipotent goddess who's been fooled by her own unintentional creation and she doesn't know it. Higher Sophia, Lain of the Wired, does know it, though. She influences Lain a little bit at a time, she peeks into her life without revealing herself. Then what happens when Lower Sophia recombines with Higher Sophia? Lain of the Wire is now able to show up in the real world with the same alien outfit representing that she's still alien to the world, but with her own face because she's reclaimed her identity. She then makes her first attempt at correcting the deficiency, but she doesn't take into consideration that Alice hasn't experienced the same Gnosis she has.

The alien is Lain herself. Her real self that the entire series is about her becoming one with, but before her real self can be recognized. First by herself, then by Alice.