As SonicRanger has pointed out, normal Lithium are pretty useless for what we are told dilithium crystals are expected to do in Star Trek. However, after 300 years of applied science and materials technology, its quite possible that a compound has been created that does things we wouldn't expect that element to be capable of. For an example from today, I'm pretty sure that people 300 years ago didn't have any idea that we would find silicon (i.e., sand, in its common form) an extremely useful, and (in final processed form) expensive material. So, yes, it might be conceivable that what is to us an exotic process combines lithium with other elements and makes a crystal. The non-canon TNG Tech manual does indicate its a compound:
http://en.wikipedia.org/wiki/Dilithium_%28Star_Trek%29
It might even be possible that lithium and dilithium crystals are created in extreme cosmic events, such as nova/supernova explosions, or by massive conventional bodies (such as failed stars, like Jupiter). The relatively desolate bodies we see it being mined on might simply be recipients of debris from some cosmic event, or the remains of a gas giant after a tremendous force has ripped away its atmosphere and forged dilithium and other elements on the surface of its core. These bodies might have been partially terraformed to permit a breathable atmosphere, and not necessarily in recent times.
The common or slang name for this this hypothetical compound could be lithium (and dilithium crystals might as well be a compound rather than merely an element of the same name, if we believe our source), because its the chief constituent element. The technological process of harvesting or creating it might be called "lithium cracking", much in the way we crack oil and reformulate it into substances we are interested in. Whether its a cheap substitute for rarer dilithium, and whether its use was discovered prior to dilithium, depends upon a lot of assumptions. It can, apparently, in a pinch be substituted for dilithium so that a starship can 'limp home' to a repair dock. And this may explain whey freighters in TOS seem to have extremely low speeds compared to 1701 (WF 2, I believe, vs. WF 6 or 8). Lithium as we know it today does, however, have some other interesting properties -- which is one indicator of why it might have been chosen in the first place and perhaps gives us reasons we might not want to assume its completely useless.
http://en.wikipedia.org/wiki/Lithium
http://en.wikipedia.org/wiki/Talk:Lithium
In terms of dilithium
just being two lithium atoms in a molecule, AFAIK atoms generally don't 'like' to exist in the singular (monoatomic), they generally are found bound together to the same element or to other elements. Yes, diatomic would be two atoms, but it seems fairly clear that the di- and tri-prefix to element names in Star Trek are being used to indicate some sort of exotic materials that (when their properties are discussed) are beyond our physical sciences today. One of the novels (I don't remember which one) came up with an explanation, something about one of the atoms being bound in warp time, or some such thing. Which is about as good an explanation as anything if you want a material 12 times harder than diamond that you can move around without an antigrav. Unfortunately Memory Beta doesn't seem to reference that novel in its expanded universe discussion of dilithium.
http://memory-beta.wikia.com/wiki/Dilithium
As to it being a slang, that might be a possible explanation, but its a fairly consistent, very briefly used slang. For reasons we all know, the show's 'shakedown cruise'. I tend to take a strong look at the expanded Trek universe and, if nothing else is available, filter continuity issues through that. Frankly, I'm a bit insulted by "ST:Enterprise" having everyone in the universe already using dilithium and antimatter to power their starships, as it contradicts "Balance of Terror". In any case I prefer to believe in gradual technological change rather than sudden changes and then centuries of relative stagnation. So, Warp 5 ships? Not a great idea in 2151, and two day trips to Kronos even worse yet.
Yes, the "stupid fans" noticed that. In "The Cage", there is clear discussion of a relatively recent break-through in propulsion technology (which 1701 is implied to have), decreasing travel time. In the "Star Trek Spaceflight Chronology" and the novel based in part on it, "The Final Reflection", dilithium's propulsion enhancing possibilities are a recently discovered property (c. 2230), courtesy of TAS's Carter Winston, that make the revolutionary Constitution class possible. So, to me -- prior to the discovery that dilithium could be cut into faceted crystals and used for focusing/concentrating energy in A/MA reactors -- a synthetic substance called Lithium crystals played a similar role, for an indeterminate time (originating sometime between ENT and TOS).
And, no, I don't particularly endorse the TNG Tech Manuals explanation that dilithium is effectively 'nature's way' of making a M/AM reactor. I prefer to believe that humanoid technology provides the magnetic fields containing the AM all the way from storage until it smashes into the matter, and the crystals then properly channel the energy from that annihilation -- but that's just me. Having the reaction take place inside the crystals at best wouldn't help the reaction along and at worst would simply blow the crystals into atoms.
Thus endeth the rant.
