您需要存储对象，而不是仅存储用于查找最小或最大值鞋的名称。
取第一项，从第二项迭代到最后一项。然后检查value是大于还是小于，并取该项。之后只返回名称。

function highestValShoe(shoeArray) {
    let shoe = shoeArray[0];
    for (let i = 1; i < shoeArray.length; i++) {
        if (shoeArray[i].valuePerItem > shoe.valuePerItem) shoe = shoeArray[i];
    }
    return shoe.name;
}

顺便说一句，请声明所有变量。如果没有，你会得到全局变量，这可能会导致有趣的结果。并使用分号。始终。

//Now I will create a class for the shoes.
class Shoes {
    constructor(name, productCode, quantity, valuePerItem) {
        this.name = name;
        this.productCode = productCode;
        this.quantity = quantity;
        this.valuePerItem = valuePerItem;
      }

    //This code will enable us to update the quantity.
    updateQuantity(newQuantity) {
        this.quantity = newQuantity;
    }
}

//This function will enable us to search for any shoe within the array.
function searchShoes(shoeName, shoeArray) {
    for (let i = 0; i < shoeArray.length; i++) {
        if (shoeArray[i].name === shoeName) return shoeArray[i];
    }
}

//This function will enable us to search for the lowest value shoe. 
function lowestValShoe(shoeArray) {
    let shoe = shoeArray[0];
    for (let i = 1; i < shoeArray.length; i++) {
        if (shoeArray[i].valuePerItem < shoe.valuePerItem) shoe = shoeArray[i];
    }
    return shoe.name;
}

function highestValShoe(shoeArray) {
    let shoe = shoeArray[0];
    for (let i = 1; i < shoeArray.length; i++) {
        if (shoeArray[i].valuePerItem > shoe.valuePerItem) shoe = shoeArray[i];
    }
    return shoe.name;
}

//Now I will create 5 instances for the class.
const
    shoeArray = [],
    converse = new Shoes("Converse", "010405", 3, 55.5),
    adidas = new Shoes("Adidas", "030602", 5, 85.0),
    nike = new Shoes("Nike", "052656", 2, 165.0),
    vans = new Shoes("Vans", "745023", 6, 95.5),
    fila = new Shoes("Fila", "034567", 3, 45.0);

//This will push all instances into the shoeArray.
shoeArray.push(converse, adidas, nike, vans, fila);

console.log(lowestValShoe(shoeArray));
console.log(highestValShoe(shoeArray));

因为你需要重写min和max函数，这样它们就不会只使用Math.min和Math.max，你可以使用Array.reduce把它们变成一行代码：
https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array/reduce

function lowestValShoe (shoeArray) {
  return shoeArray.reduce((prev, curr) => (prev.valuePerItem < curr.valuePerItem ? prev : curr));
}

function highestValShoe (shoeArray){
   return shoeArray.reduce((prev, curr) => (prev.valuePerItem > curr.valuePerItem ? prev : curr));   
}

在这里，您可以看到它实时返回Fila和Nike，分别表示最小值和最大值：

//First I will establish an empty array to push() the shoes in later.
shoeArray = [];

//Now I will create a class for the shoes.
class Shoes {
  constructor(name, productCode, quantity, valuePerItem) {
    this.name = name;
    this.productCode = productCode;
    this.quantity = quantity;
    this.valuePerItem = valuePerItem;
  }

  //This code will enable us to update the quantity.
  updateQuantity(newQuantity) {
    this.quantity = newQuantity;
  }
}

//Now I will create 5 instances for the class.
converse = new Shoes("Converse", 010405, 3, 55.5);
adidas = new Shoes("Adidas", 030602, 5, 85.0);
nike = new Shoes("Nike", 052656, 2, 165.0);
vans = new Shoes("Vans", 745023, 6, 95.5);
fila = new Shoes("Fila", 034567, 3, 45.0);

//This will push all instances into the shoeArray.
shoeArray.push(converse, adidas, nike, vans, fila);

console.log(lowestValShoe(shoeArray));
console.log(highestValShoe(shoeArray));

function lowestValShoe(shoeArray) {
  return shoeArray.reduce((prev, curr) => (prev.valuePerItem < curr.valuePerItem ? prev : curr));
}

function highestValShoe(shoeArray){
   return shoeArray.reduce((prev, curr) => (prev.valuePerItem > curr.valuePerItem ? prev : curr));   
}