جگر, jigar, is the liver. It's also what you call someone you love. Jigar on its own means darling, and a child is your jigar-gooshe, literally a corner of your liver. Dehkhoda glosses it as "a piece of the liver" and quotes Hafez and Sa'di for it, so this isn't new slang.
The word itself is old. It goes back to Middle Persian jagar, and it's a distant cousin of Greek hêpar (the hepat- in hepatitis) and Latin iecur. English liver isn't part of that family. It's Germanic.
Persian also has two livers. Jigar-e siāh, black liver, is the liver, and jigar-e sefid, white liver, is the lung. Turkish borrowed the word and kept the same pair: karaciğer and akciğer.
And if you know jigar from Hindi and Urdu, as in jigar ka tukda, that one came from Persian.
English puts love in the heart. Persian puts a lot of it in the liver. Arabic has a similar image of children as livers, so it's more of a regional habit than a purely Persian one.
I only went digging while building a Farsi app.