From the WP article about Castle Bravo:
a theoretical error made by designers... They considered only the lithium-6 isotope in the lithium deuteride secondary to be reactive; the lithium-7 isotope, accounting for 60% of the lithium content, was assumed to be inert. ... It was assumed that the lithium-7 would absorb one neutron, producing lithium-8 which decays (via beryllium-8) to a pair of alpha particles on a timescale of seconds—vastly longer than the timescale of nuclear detonation. However, when lithium-7 is bombarded with energetic neutrons, rather than simply absorbing a neutron, it captures the neutron and decays almost instantly into an alpha particle, a tritium nucleus, and another neutron.
The result was an extra 10 megatons of yield (4.2e16 joules, i.e. 42 petajoules) from the 10.7 tonnes of the device, most of which was presumably the lithium; this is about 6 or 7 petajoules per kg.
This reaction is remarkably similar to the “energy amplification” reaction used in thorium reactors: a neutron entering a mass of lithium-7 will stimulate a decay, which emits another neutron, which is either lost or stimulates another decay, and so on. We can infer that this does not produce a self-sustaining chain reaction because natural lithium still contains 92.5% lithium-7, and there is apparently no critical mass of lithium which results in a uranium-like or plutonium-like chain reaction.
(Presumably this is because in fact the decay to two helium nuclei via ⁸Be mentioned above does happen even at high energies and consumes some or most of the neutrons. This was in fact the first artificial “splitting of the atom”, carried out by Cockcroft and Walton in 1932 with a 700kV tube to accelerate protons — the cyclotron, necessary to reach energies per charge higher than your voltage, was only invented that same year in Berkeley.)
Lithium, however, is more appealing than thorium for a variety of reasons, occurs at 17 ppm in the crust, compared to thorium’s 6 ppm, and there is an existing industry extracting over half a million tonnes of it from the crust per year, while thorium only has a few uses and is somewhat dangerous to refine because of its natural radioactivity. Thorium is imported into the US at a few tens of tonnes per year, and costs around US$100 per kilogram, while lithium is imported at a rate of some 3000 tonnes per year and costs around US$6000 per tonne, or US$6 per kilogram.
Lithium-6 also produces tritium when irradiated with a neutron, yielding 4.8 MeV (about 40 times less than the energy of an actinide fission), which works out to 77 TJ/kg, about 1% of the energy density inferred above for the Castle Bravo excess energy.
(The World Nuclear Association says that lithium-7 constitutes 92.5% of natural lithium and has a very low neutron cross section of 0.045 barns, and that on proton bombardment it fissions to 2He-4 yielding 17 MeV.)
US$6/kg and 6 PJ/kg conveniently give a fuel cost of US$1/PJ. By comparison, a common wholesale price for electrical energy is US$60/MWh (though this fluctuates minute by minute and goes negative most nights). This electrical price works out to US$16.67 per gigajoule, and thus US$16'666'667 per petajoule, some three million times the price of the lithium fast fission energy. Even the lower energy of the ⁶Li decay would be some thirty thousand times cheaper than grid power.
The tritium produced also decays energetically with a half-life of some 12 years; while this is too slow to be useful for a nuclear weapon, it could be useful for a nuclear reactor.
There’s still an engineering question about how difficult it is to generate the neutrons to initiate the reaction; fusing lithium deuteride to generate the neutrons is clearly not an option in most cases, and you need to take into account the energy consumed by the particle accelerator. I suspect that the slow fission reaction by way of ⁸Be mentioned above would also work, in which case the neutrons need not be energetic. Quite likely the usual approach of neutron spallation from a mercury target impacted by a proton beam would be adequate.
Lithium is particularly troublesome here because of its small cross-section — while ²³⁸U has a thermal neutron capture cross-section of about 2 barns, I’m guessing both ⁶Li and ⁷Li have capture and fission cross-sections in the neighborhood of the capture cross section of hydrogen (0.2 barns), deuterium (0.0003 barns), or carbon (0.002 barns), which are another two to four orders of magnitude lower for fast neutrons. This means you might need 10 or 100 or 1000 times as much thickness of lithium absorbing the neutrons as you would for an actinide, or, alternatively, 10 or 100 or 1000 times as many neutrons.
The fission produces helium, a gas, and tritium, which can be gaseous if it reacts with other tritium rather than something else in the area; this suggests that the fuel should be kept liquid, perhaps as a molten salt, to allow the gas to bubble out instead of building up inside a solid.
Given the uniquely prominent position of ⁷Li/⁸Be fission in the history of nuclear physics research, it is inconceivable that this idea is original, so there is probably an obvious, well-known reason why it doesn’t work; Szilárd surely tried it in the early 1930s, ten years before he and Fermi got a successful chain reaction going in uranium in 1942, but of course Szilárd didn’t have access to modern fast electronics or even a cyclotron at the time.