Update the attribute value of an object using the map function in ES6

try this, ES6 Object.assign() to create copy of array element and update new object.

let schools = [{
        name: 'YorkTown',
        country: 'Spain'
    },
    {
        name: 'Stanford',
        country: 'USA'
    },
    {
        name: 'Gymnasium Achern',
        country: 'Germany'
    }
];

const editSchoolName = (schools, oldName, name) => {
    return schools.map(item => {
        var temp = Object.assign({}, item);
        if (temp.name === oldName) {
            temp.name = name;
        }
        return temp;
    });
}

var updatedSchools = editSchoolName(schools, "YorkTown", "New Gen");
console.log(updatedSchools);
console.log(schools);

Using destructuring

const schools = [
  {
    name: "YorkTown",
    country: "Spain",
  },
  {
    name: "Stanford",
    country: "USA",
  },
  {
    name: "Gymnasium Achern",
    country: "Germany",
  },
];
const editSchoolName = (schools, oldName, newName) =>
  schools.map(({ name, ...school }) => ({
    ...school,
    name: oldName === name ? newName : name,
  }));
const updatedSchools = editSchoolName(schools, "YorkTown", "New Gen");
console.log(updatedSchools);

You need to return the updated object:

const editSchoolName = (schools, oldName, name) =>
  schools.map(item => {
      if (item.name === oldName) {
        return {...item, name};
      } else {
        return item;
      }
});

   const editSchoolName = (schools, oldName, newName) =>
    schools.map(({name, ...school }) => ({ ...school, name: oldName === name ? newName : name }));

You could shorten it by using a ternary.