[c#] Best way to update an element in a generic List

Suppose we have a class called Dog with two strings "Name" and "Id". Now suppose we have a list with 4 dogs in it. If you wanted to change the name of the Dog with the "Id" of "2" what would be the best way to do it?

Dog d1 = new Dog("Fluffy", "1");
Dog d2 = new Dog("Rex", "2");
Dog d3 = new Dog("Luna", "3");
Dog d4 = new Dog("Willie", "4");

List<Dog> AllDogs = new List<Dog>()
AllDogs.Add(d1);
AllDogs.Add(d2);
AllDogs.Add(d3);
AllDogs.Add(d4);

You could do:

var matchingDog = AllDogs.FirstOrDefault(dog => dog.Id == "2"));

This will return the matching dog, else it will return null.

You can then set the property like follows:

if (matchingDog != null)
    matchingDog.Name = "New Dog Name";

If the list is sorted (as happens to be in the example) a binary search on index certainly works.

    public static Dog Find(List<Dog> AllDogs, string Id)
    {
        int p = 0;
        int n = AllDogs.Count;
        while (true)
        {
            int m = (n + p) / 2;
            Dog d = AllDogs[m];
            int r = string.Compare(Id, d.Id);
            if (r == 0)
                return d;
            if (m == p)
                return null;
            if (r < 0)
                n = m;
            if (r > 0)
                p = m;
        }
    }

Not sure what the LINQ version of this would be.