How on earth does Leona have anything passable as "AD" other than her Q?
Anyway, regarding AP Leona, me and Ikaru agreed that it was wouldn't work out unless you had pre-co-ordinated with your team in which case it could actually be decently powerful.
The main problem is that AP Leona wouldn't fit the definition of a burster or anything so she wouldn't be a role-centralizing pick so much as a team centralizing one. Aside from Q+Lich Bane and her Ult, she has no stable form of good damage (though Q is good regardless).
So if this is the case, it isn't gonna stop Leona from using CC and stuff and still securing kills for her team. If she's building AP, though, there's a few key differences between her and normal Leona:
Farming of minions and possible kills lead to a higher gold income, letting Leona buy more expensive items which would benefit her towards her intended purpose (e.g Iceborn Gauntlet).
The damage combined with her CC leave her more of a case of "here they're exposed and damaged for about like 40% of their health we can kill them" rather than "here they're exposed try to kill them".
Co-ordinating with ganks with the jungler is more key to your survival in the game so you can keep up and secure kills early on.
So, since AP Leona wouldn't be full burst, the team would require a offense-focused jungler or top laner aswell to make up for the lost damage; or just a 3rd source of (half-ish) burst damage in general. Also, AP Leona most likely wouldn't build items like talisman of ascension or passive buff items; her build contributes to teamfight effectiveness in different ways than a support leona would; Rylai's slow over Locket buff, the like. It's arguable that the build pros of AP Leona might not be as worthwhile if your team isn't building items that can work with it. Of course, that's basic LoL logic, but AP Leona would be particularly vulnerable to it.
The problem is, without the team co-ordinating, picking their champs and their builds specifically, Leona's purpose is weakened because she can't dish out all the AP burst on her own and, like with any Leona, she wouldn't be able to execute disruption in teamfights as effectively; Leona is more affected by team co-ordination than most champions. And in normal games with random people, people won't have planned with you before the game and therefore AP Leona would lose presence.
There's also the side note of her having some decent bulk, but like Ame said if you're looking for a tanky AP mid you're better off with Malphite or the like, so that's more of a minor pro.
So, ultimately:
AP Leona's main attraction over normal Leona is the slight damage added on top to the disruption already there.
AP Leona's possible kill securing and actual farm would lead to her buying more expensive items that contribute to the team in different ways, such as Abyssal Scepter's magic resist debuff or the slows of Rylai's/Iceborn.
She's weak early/early-mid game on her own since her damage wouldn't be as high as the enemy mid laner's, if we're to assume the enemy is picking the standard mid laner. However, her CC disruption and lackluster/decent damage would work well when co-ordinating in a gank, especially if the ganker is also CC heavy (think Nautilus, Rammus, Blitzcrank, etc).
Late game, her Q is her best damage/overall use tool due to items like Lich Bane and Iceborn Gauntlet's contributions and the stun it provides, along with whatever other on-hit effects you may have. Her ult can do insane damage, but it has a large cooldown. Overall, Leona won't have as much burst presence as the usual AP carry would.
Personal opinion, Leona would be more of a counterpicker to certain champions rather than a standard AP mid, due to how her disruption would work; of course, if the enemy specializes in harassment then Leona won't be able to retaliate easily and she's screwed instead (e.g Ziggs, possibly Karthus).
In a premade game where she's co-ordinated with her team, she can make the best use of her disruption and decent damage to fuck over the enemy team. This ultimately means she's more of a damage-support than a burst carry, especially with things like Sunlight. You can also pull off some fancy combo shit if you like; say, if you have a Rumble at top to make up for some lack of Leona burst then you can double ult and he'll Q them or something. Experimentation and the like.
So yeah, it's not really a viable strategy in normal gaming. Leona's ultimately more prone to the effect of teamwork, AP or not, and without a proper plan with you're team you're pretty much fucked. However, if you've actually planned shit out with a premade team or something, then she can actually be pretty damn threatening. That applies to almost anything in LoL, but as I said, Leona's particularly prone to it.
In short: Don't do it with strangers. It's good to do it with premades where you've planned it, though. Ultimately your choice, and pretty much something you're barely ever going to see.
I think I'm gonna want to give it a try with some of my friends sometime. This is essentially all theorycrafting, as a disclaimer; I don't have Leona yet, so I can't really speak too much for it.