A lavish party and a rainstorm have the same ancestor. The word began as a name for a downpour.
In French, lavasse or lavache meant a flood of rain. Etymonline dates the noun to the 15th century and calls it Old French. Dictionary.com labels the same forms Middle French, so the period is a matter of where one draws the line. Both agree on the source: the verb laver, "to wash," from Latin lavare, which also means "to wash."
Middle English borrowed the idea as lavas. Dictionary.com glosses it as "profusion" as a noun and "profuse" as an adjective. The picture behind it is easy to see. Something that pours down in a torrent comes in quantity, without measure.
By the mid-1400s the adjective lavish meant "spending or bestowing profusely." It described a giver as much as a gift. The verb, to lavish something on someone, was formed from the adjective in the 1540s. Dictionary.com's usage note captures the tone: lavish suggests generosity that can tip into excess, while prodigal suggests plain waste.
What makes the word unusual is that the water dropped out of it completely. Nobody hearing "lavish" today thinks of rain. Yet the image of a shower of good things is still built into how the word is used. Praise, attention and money are all lavished, poured out on someone.
The washing root has a large family in English. Etymonline traces Latin lavare to an ancient root, leue-, "to wash." Lavatory began as a washbasin in the late 1300s and became a polite word for a toilet in 1864. Laundry came from Old French lavanderie, a wash-house. The old verb lave, "to wash by pouring water," is first recorded around 1200. Deluge, ablution and lotion belong to the same family.