Peculiar Cocoons are the raw materials that you encounter in your journey in Wild Hearts. This material can be used to make smooth silk which can then be used in the crafting of weapons and armor that will be useful later in the game.
Peculiar Cocoons are white and often found in the same locations as the other cocoons from which you get Silken Thread. Their appearance is that of medium-sized balls clumped together and standing up not more than 3 feet with blue cotton buds-like protrusions poking out on all sides.
Wild Hearts Peculiar Cocoon locations
Peculiar Cocoons are not available at the start of the game but they appear after you have played the game quite a bit. When you reach Chapter 2 of the game, some new gear will be unlocked as well as a new location known as the Fuyufusagi Fort will be unlocked.
The Fuyufusagi fort is a ruined town which the Peculiar Cocoons have made their home. The Cocoons are white so it becomes hard to find them here because the whole area is covered with snow. The Peculiar Cocoons can be found all around the Fort as well as in it.

Best way to farm Peculiar Cocoons in Wild Hearts
Now, as you may need more than three Cocoons to use them in the crafting of armor and weapons, you will need more. An easy way is to look at the place where they will spawn more frequently.
Now near the top of the main fort, you will find an Ancient Well. Near this well, there is a sure chance that these cocoons will spawn. You can use this to your advantage by placing a Camp Site here.
Now when you open the map and look for Kemonos to hunt in the Fuyufusagi Fort, defeat them and then come back. When you come back to the Camp Site you will find that the Cocoons are reloaded on their spawn locations near the Well.
Another way to farm them is to use the Tsukumo Ore Shrines at any camp. This will collect random resources native to this region as time passes. Thus you can collect more cocoons as time passes. Using them both will give you tons of Cocoons to use in Crafting Armour and Weapons.