[{"name":"parent 1","children":[{"name":"child 1","_id":ObjectId("63a94a4a1baa3d5bfdc22fb4")},{"name":"child 2","_id":ObjectId("63a94a4a1baa3d5bfdc22fb4")}]}]

db.collection.update({"name":"parent 1","children._id":ObjectId("63a94a4a1baa3d5bfdc22fb4")},{"$set":{"children.$.name":"new name"}})