Amelia is like no other female character we've gotten on this show. She is (one fervently hopes) far better with animals than with people. She's crusty, a grieving widow, quite possibly a burgeoning alcoholic. She has baggage. She has zero interest in providing emotional support to anyone. She is no one's nurse maid (and thank goodness for that, because her bedside manner would be terrible). She's running away from home, family, and whatever emotional support she might have found there.
She is, in her own words, a mess. She may be the messiest female character we've ever had on the show; certainly she's the one whose issues have received the longest sustained focus.
And you see, she comes into the picture just in time to meet Sam, who is also a mess, also running and grieving and barely functional by any definition of the word. But she doesn't immediately like him. She is entirely unimpressed by his charms, his arms, his wardrobe, and even his plumbing skills. Despite all that, she and Sam manage to find comfort in each other that they haven't been able to (or perhaps haven't dared try to) find in anyone else. They are two broken people guarding themselves very carefully, even from each other, but that equity between them is a lot of what makes them work. They both need the same things: physical comfort, another body bumping around the house to keep them from being alone, someone to give a damn but not ask too many questions, someone who's experienced deep grief.
They meet as equals, which Winchesters rarely ever do with other characters, particularly female civilians. Amelia does not need rescuing from some supernatural baddie, nor is she shown up by Sam's hunting expertise. And yet, even as a female civilian, she is given development and complexity of her own; narratively, Amelia is still serving Sam's story (via, it must be said, the most horrid soft focus known to network TV), but the show doesn't use that as an excuse to refrain from giving her her own complex development. She is not Jess, alive for one episode and serving forever after as a symbol of Sam's grief. She is not Lisa, an extremely generous and warm-hearted woman who welcomed into her home and her life a man she'd met only a handful of times, very conveniently for both him and the narrative; who somehow remained a cipher despite appearing on the show (alive!) in eleven different episodes. Amelia is the Lisa antithesis: she is supremely inconvenient.
(Interestingly, I think the only other civilian girlfriend character who comes close to receiving Amelia's complexity of motivation and relationship with the Winchesters is Cassie Robinson, although with only one episode to work with, we can only begin to glimpse all her layers.)
What's more, unlike all the other civilian girlfriend characters, the Sam/Amelia relationship is developed on screen. IMO the single biggest flaw in the execution of Dean/Lisa is the fact that all the key aspects of their development happen offscreen. Either that, or there were no other developments, which would be even worse. It drives me batty the way canon just seems to assume that of course Lisa would take him in, of course they would settle into a comfortable if bittersweet romantic/sexual relationship in that year Sam was gone. THERE IS NO OF COURSE HERE. Ahem. But Sam/Amelia gets all its key moments on screen, and that makes a world of difference. And they're tentative, halting moments, two steps forward and one step back, and that pleases me, too; I will always prefer complicated to simple, and messy to instalove.
This is not to say the execution is perfect. The soft-focus filming of all their scenes is peculiar at best, their arc goes from heartfelt to silly with Don's soap operatic resurrection, and their last flashback ep together is full of trite, ridiculous dialogue and overwrought melodrama that's completely out of place given the kind of relationship they had prior. I can wank explanations, them clinging to a false idea of their relationship as security in lives that turning upside down yet again, but I can't quite sell myself on it, and I can't like it.
Still. I adore them, and I adore her. In purely superficial terms, I love that Amelia dresses so casually, in peasant blouses and oversized button downs, that she has curly hair, that she hasn't been pressed and shined to look like every other female character on the CW. (If you ever want to send me into instarage, whine about how much better she'd look if she straightened her hair. Just try it.) Also, Liane Baliban is gorgeous. In emotional terms, Amelia breaks my heart, alone and with Sam, and she is a character type I rarely seen, filling a hole I didn't realize I needed filled.
I love them. I love her. The end.