Budding Prismarine
Grows Prismarine Crystal Shards, the way Budding Amethyst grows amethyst.
Finding It
The easiest place to find it is the Prismarine Reef, where about every other chunk has some in the top of the sea floor and in the prismarine patches. It also forms in the top of the sea floor of the Abyssal Trench and the Bioluminescent Grove, and in the floor of every Glow Grotto. Vanilla Warm Oceans and Deep Lukewarm Oceans have the odd one in the top of the sea floor too. Fishing in a lucky zone in the Abyssal Trench can also bring one up as treasure. Mine it with Silk Touch and place it wherever you want to farm it.
The block gives off a faint glow and the odd sparkle, so look for a dim light on the sea floor, which is easiest at night. To reach the right biome, set an 
Eye of the Sea to Prismarine Reefs and throw it.
A 
Grafting Tool scrapes Lithophiles off it in any of those biomes.
Growth Stages
Buds sprout on any face that has air or a water source block next to it, so a block on the sea floor grows just as well as one in a cave. Each bud grows in place through four stages:
Small Prismarine Bud, Medium Prismarine Bud, Large Prismarine Bud, then Prismarine Cluster. Buds glow brighter as they grow.
Harvesting
Wait for the Prismarine Cluster. Breaking one drops two 
Prismarine Crystal Shards, more with Fortune, and a new bud starts in its place later.
Buds broken early drop nothing unless you use Silk Touch, which picks up the bud or cluster itself. Harvest only what grows on the Budding Prismarine block.
To move the block itself, mine it with a Silk Touch pickaxe. Mined with any other pickaxe it breaks into 4 to 6 Prismarine Crystal Shards and stops growing anything.