Replace a object in a list of objects

You have to replace the item, not the value of customListItem2. Just replace following:

customListItem2 = customListItems.Where(i=> i.name == "Item 2").First();
customListItem2 = newCustomListItem;

With this:

customListItem2 = customListItems.Where(i=> i.name == "Item 2").First();
var index = customListItems.IndexOf(customListItem2);

if(index != -1)
    customListItems[index] = newCustomListItem;

Edit:

As Roman R. stated in a comment, you can replace the .Where(predicate).First() by a simple First(predicate):

customListItem2 = customListItems.First(i=> i.name == "Item 2");

var customListItems = new List<CustomListItem>();
var customListItem1 = new CustomListItem() { name = "Item 1", date = DateTime.MinValue };
var customListItem2 = new CustomListItem() { name = "Item 2", date = DateTime.MinValue };
var customListItem3 = new CustomListItem() { name = "Item 3", date = DateTime.MinValue };

customListItems.Add(customListItem1);
customListItems.Add(customListItem2);
customListItems.Add(customListItem3);

var newCustomListItem = new CustomListItem() { name = "Item 4", date = DateTime.Now };

customListItems[customListItems.FindIndex(x => x.name == "Item 2")] = newCustomListItem;

or

public static class ListExtensions
{
    public static void Replace<T>(this List<T> list, Predicate<T> oldItemSelector , T newItem)
    {
        //check for different situations here and throw exception
        //if list contains multiple items that match the predicate
        //or check for nullability of list and etc ...
        var oldItemIndex = list.FindIndex(oldItemSelector);
        list[oldItemIndex] = newItem;
    }
}

and then

customListItems.Replace(x => x.name == "Item 2", newCustomListItem);