The word is not one of that melancholy collection ending in -cide that refers to an act of killing or something that kills (suicide, pesticide), since it comes from a different Latin verb, caedere, to fall. The first part is from Latin stilla, a drop; the English word is a reformulation of Latin stillicidium, falling drops.
The Latin word could mean in particular the drip of rain from the eaves of a house, which is exactly equivalent to an ancient meaning of our eavesdrop. This meaning led to the main historical sense of the word, a legal term in Scots law. If a householder let rain fall from his eaves on to the land of a neighbour, he needed the neighbour’s permission. John Erskine explained this in 1754 in his Principles of the Law of Scotland: “No proprietor can build, so as to throw the rain water falling from his own house immediately upon his neighbour’s ground, without a special servitude, which is called of stillicide.” (Servitude is another term for easement, a special permission relating to access to land.)
It’s not a word much encountered these days. When it appears it has the sense of falling water, not the legal one. It is in a poem in Vladimir Nabokov’s Pale Fire: “Stilettos of a frozen stillicide”, one of a collection of unusual words in that section that also includes shagbark, torquated, vermiculated, preterist, iridule, and lemniscate. Its most famous use is perhaps that by Thomas Hardy, again in a poem:
They’ve a way of whispering to me —
fellow-wight who yet abide —
In the muted, measured note
Of a ripple under archways,
or a lone cave’s stillicide.