The Jar of Yummy Pickles is an item that is given to you by Mixolydia to give to Ionia. Lucas will drop this item on Argilla Pass, and you will have to search for it. While you are looking for them, Boney will also find other various items. You cannot eat these, but only give them away.
The only game that this actually plays a major role in is Mother 3. It doesn't have anything else to do with any other game.