Beyond what is mentioned above, another reason lithium deuteride was (perhaps still is) used in thermonuclear weapons is that in addition to the thermonuclear fusion reaction of deuterium alone, there is a strongly exothermic nuclear reaction between lithium6 and deuterium as well. The NRL Plasma Physics Formulary says the fusion of these nuclei yields a 22 MeV photon, quite a lot of energy per nucleon.

The Mike device tested in November of 1952 used cryogenic liquid hydrogen as the thermonuclear fuel source. However, the test device weighed several tons and couldn't possibly be delivered via bomber or missile because of the required cryogenics system for the liquid hydrogen. So when the United States started weaponizing thermonuclear explosive designs in the 1950's, they switched from liquid deuterium and tritium to the dry (at room temperature) and chemically stable lithium deuteride.

Richard Rhodes noted in Dark Sun: The Making of the Hydrogen Bomb, the Operation Castle tests were the first weapons tests of high-yield hydrogen bombs using lithium deuteride as the secondary thermonuclear fuel. According to Rhodes, the designers only took into account the reaction between lithium6 and deuterium when they calculated the yield, but apparently there is a significant additional reaction where lithium7 fissions to produce a tritium atom, and an alpha particle. The tritium then contributes to the reaction. As a result, both the Castle Bravo and Castle Romeo tests had yields significantly higher than expected -- though the true yield is classified, Bravo probably had a yield around 15 megatons, by accident the largest nuclear test conducted by the United States.) The Bravo test was responsible not only for the contamination of Bikini Atoll, but also for the contamination of the Japanese fishing vessel Diago Fukuryu Maru (Fifth Lucky Dragon) on which several fishermen contracted radiation sickness and one died. The latter caused a huge anti-US uproar in Japan throughout 1954.

Sources: Dark Sun by Richard Rhodes