Figured I'd contribute my thoughts to your A/L project as that pairing was the first one I wrote when I began writing in the LOTR fandom in October, 2002. It's funny because I hadn't read any fics with that pairing prior to writing Ohtarnil. It just seemed a natural one to write and others seem to have agreed as I believe it is the most popular pairing in the fandom.
I think the main reason for this is the beauty of both actors who portrayed Aragorn and Legolas in the movie. This and the clear chemistry between them is what makes Orlando/Viggo the most popular RPS combo as well. There were a number of scenes between Aragorn and Legolas in the movie that just sizzle, for example, Legolas's emotional defense of Aragorn at the Council of Elrond, and another, what Capella calls "the lovers' tiff" at Helm's Deep, the scene where Legolas brings Aragorn his sword and asks his forgiveness, which certainly has the feel of a reconciliation of a lovers' quarrel. In fact, I'd say the chemistry was so apparent even to PJ that they didn't dare have Aragorn hug Legolas in the famous "you're late" scene. Think about it. Aragorn hugs Gimli and later Haldir. Why not Legs? Well, because it would have been JUST TOO MUCH. LOL. The wedding scene in ROTK seems to feed into this (I wonder if by then PJ wasn't catering to this underground fandom) because Legolas certainly comes to Aragorn with the look of a bride, stunning in white.
In addition, the characters seem well-matched. Aragorn and Legolas are both protectors of their respective peoples, both noble, fierce warriors with a sense of duty. And Aragorn, although a man, has been raised by elves and has studied elvish lore. Tolkien portrays him as a true man of Numenor, the race that blended elves and men. Aragorn represents the best qualities of that combination. This makes for a natural attraction between him and Legolas. In addition, Aragorn has a heavy burden, seems moody and uncertain about his fate, which creates a supportive role for Legolas, especially once Gandalf "dies" in Moria and Legs becomes Aragorn's lieutenant for the rest of the journey. Certainly, it is clear, both in Tolkien's book and in the movie that they become good friends.
I think the main factor of their appeal is the sense of limited time, "the few stolen moments" that brings a special urgency to their relationship. The urgency is both romantic and highly erotic. There are two aspects of this. The first is that they are of different races, one immortal and one mortal, so ultimately they will be separated by death. The terrible sadness of the ending of the Elder days manifested by the elves leaving Middle Earth is a dominant theme in Tolkien's LOTR which finds an echo in Araleggo angst. The second is the canon problem of Aragorn's marriage to Arwen, which usually means that the affair with Leggy must end. Both of these factors create an angsty tension-filled relationship that lends itself to story. And I would guess that 90% of the fics revolve around these issues.
What to do about Arwen? If one cares about remaining true to Tolkien canon (and I've seen fics that blithely ignore her) then one has to reconcile this issue. I've seen various approaches to this, ranging from Arwen is merely a cold marriage of convenience because Aragorn truly loves Legolas, to Aragorn is torn equally in his affections, to the kindest and most creative solution in Esmeralda's fics in which Arwen is a full participant in the relationship. I actually prefer Esmeralda's approach as too often I see Arwen villified in fanfic. It smacks too much of women-bashing to my taste (but that's an aside).
I came to write Ohtarnil through contemplation of the "what to do about Arwen" issue. How to get these gorgeous guys with such natural chemistry together and remain true to their personalities and canon? My solution drew on what I knew of ancient Greek warrior tradition which allowed, and in fact encouraged, homosexual relationships as a form of bonding. It makes sense as it creates a culturally sanctioned allowance for men's sexual needs in the absence of women, and emotional bonds that make the men all the more likely to protect one another in battle. Witness Achilles and Patroclus. So, I invented a similar cultural idea for the elves, that of the Ohtarnil, a warrior-love. This allowed Aragorn to have a "safe" space in which he and Legolas could have a relationship. Unfortunately, under this solution, once the war was over and there was no more need for the warrior-bond, it would have to end, as happened in my fic, "Dark to Dawn," in which Legolas despairs on Aragorn's wedding night.
Anyway, those are a few of my thoughts on the pairing.
And I've now gone overlong with this post. Hope it was somewhat helpful to you.