Lina X Amelia 4-Ever!
Let's face it, in nearly any series there can be some form of evidence, no matter how obscure to support any pairing you can think of. And even if there isn't by some quirk of fate, if you look at any relationship from the right angle, even a relationship based entirely on hate and malice, you can say that the people are just denying their feelings for one another. Couple that with the increasing acceptance of same sex relationships and you come to realize that just a handful of excuses can be used to justify any relationship you can think up!
But I digress, what I was trying to say before was simply, all reasoning for any relationship can be based on any piece of information if it is spun the right way. That being said, it is still impossible for me to make an argument for Lina x Amelia that will hold any water, there will always be holes to be punched in any argument for any relationship be it canon or simply the male lesbian fantasy over-exerting itself.
First, what they have in common. Despite the notion that opposites attract, many fruitful relationships have some solid common ground. (Note that some of these points are so intuitive that to re-state them might infuriate some long term fans, I will do so anyway to pad the list a bit.)
Similar Ages : Although in the anime Amelia is made noticeably younger then Lina in the Novels they are depicted as being the same age.
Sorceresses : Despite their training in opposite fields of magic, Amelia and Lina both are practitioners of the mystical arts, additionally they have an overlapping range of magic so although they specialize different aspects of magic they have some common ground.
Temperament : Lina may be high strung and a little greedy as a baseline vs. Amelia's propensity for generosity and righteousness, but when they both really want something (Usually Food) They're damn near identical.
Intelligence : In the anime Amelia is a little slower then Lina when it comes to intelligence, but in the Novels they are very much on equal footing. Equal intelligence can be an excellent base for most relationships. That is one of the points that people point out against Gourry, that Lina seems to get frustrated with his inability to comprehend simple points. Equal intelligence means meaningful fulfilling conversations.
Of course, going against what I said above, there are instances when opposites do attract. It is good to have differences to increase the interest between to parties but what matters is having the common ground as well. That established, here are some points that compliment Lina and Amelia.
Magic : Expanding on the previous point, they do practice different sets of magic, not only that but they have notably different magical backgrounds, specifically their training. They have the ability to help one another grow in new ways.
Background : This has all the makings of the Prince and the Pauper type story, Amelia having the royal background and Lina almost falling into the category of Ronin. Although Lina might act petty enough at times, money would likely not be a major motivating factor toward her falling for Amelia although it might be something that she might consider jokingly.
Beyond the simple points above, there is much more to take into account. Aside from hints given in the Anime, Manga, and Novels that might be pointed out sequentially, there are some larger aspects to consider.
Hero Worship : A favorite theme of mine in fanfiction. Amelia does in fact idolize Lina, though the intensity of this varies from incredibly promenade shortly after they first meet up to more manageable later in the series. This kind of fanatical obsession with Lina is more in tune with Amelia's personality. Amelia is an intense person, and that intensity, the way it drops off, it's not as if she starts to dislike Lina or anything, instead her Hero worship is replaced by a sort of caring and appreciation like you would expect between two good friends. Although her relationship with Zelgadiss and Gourry (If you would call that any kind of interaction) seems to remain stagnant though the series. It's easy to see an idol become a target of love though or at the least misplaced affection as is the case with many teenagers idolizing movie stars and singers.
Lina's Temperament toward Amelia : Lina is a reserved person when it comes to showing anyone any attention other then her usual casual attitude or anger. But there are two people who are prone to getting warmth from her, Sylphiel (who gets treated different by Lina then anyone in the series) and Amelia (and I guess to a lesser extent Gourry). The picture at the top of the page shows a scene from Lina's fight with Gaav where she ends up injured, the look on her face, she is savoring that hug, if it had been from anyone else... well... it's debatable but I personally don't think it would be as pleasant in the end for both of them.
What I have been trying to establish up to this point is not necessarily that Lina and Amelia are in love and there is some conspiracy in the show to cover it up, but that they are in fact good friends. Something that many people would not have argued with me to begin with. I'm not trying to convince my readers that them being in love is what the show is trying to point out, simply that it is not something that could be ruled out with 100% certainly. After all, friends do make some of the best lovers.
As for their personal opinions on same sex relationships. Those have to be inferred from what is shown in the series. One of the most poignant events that I can think of relating to Lina comes from the Slayers Special Manga. In one of the manga installments a wealthy countess drugs Lina with the intention of having her way with her. She is unable to defend herself, but she isn't put off by the affection, more angry at the way it was garnered. When she first meets with Sylphiel in the novels she says that if she were a man (Lina) that she would have fallen in love with Sylphiel on the spot. More recently in the Slayers Neo manga there is a scene where Lina and Amelia are sleeping together and Lina is feeling up Amelia as they sleep. As for Amelia, she has made it clear that 'True Love' is the ultimate ambition and that nothing can come between it. A significant example of this is in Slayers TRY where Amelia is stranded on a island consisting of fish people with a few humans and she defends a relationship between a human and a fish person. One could argue that this belief in choosing ones partner might not extend to same sex relationships, be it through religious or personal beliefs, then again you could argue in the opposite direction as well.
The critical reader though (however many I have left by this point) might realize that although I have been giving reasoning behind Lina and Amelia as a couple as taken from their behaviors and actions, I have yet to give my personal reasoning behind the coupling. Well, there are actually two reasons behind the pairing. 1) When I first got into the fandom I didn't see anything Yuri so I opted to write something 2) Their relationship would be very, very, spicy. Think about it, Lina being such a strong willed person and Amelia such a shut-in innocent... very nice [+3) They look cute together].
I hope this encourages more people out there to write more Lina x Amelia fanfiction. I've written a lot for this pair and over the years I have kind of moved away from it and more toward the traditional Lina x Gourry and Zelgadiss x Amelia pairings but it still holds a special place in my fandom. At one time I had a shrine to Lina and Amelia but for some reason anglefire decided to take it down, oh well, can't win em' all.