diamonds sound close to adamant and both are touted for being the hardest substance
adamant goes to Saturn's sickle and more importantly adamantine prison, place where fallen angel watchers are kept
Lucy in the Sky with Diamonds could be worked out as Lucifer in the Sky (i.e. Saturn, the black sun that gives invisible light leading to illumination) with the diamonds either being shining stars (angels) or harkening back to adamant therefore pointing to Saturn as the place of or symbol for adamantine prison (compare to Alice in Chains - Alice=Lucy=Lucifer and chains being the chains of adamant)
there's always more to this stuff but that's the best explanation I've found so far