I am trying to add an additional option to the looping menu that will allow the user to select a Student, then display the full details for just that Student

Normally we do not do your homework.
But you showed us what you already achieved.
Program.cs

using System;
using System.Collections.Generic;
using System.Linq;

namespace LM
{
    class Program
    {
        static void Main(string[] args)
        {
            var students = new List<Student>();

            var newStudent1 = new Student();
            newStudent1.Id = 0;
            newStudent1.Name = "Bob";
            newStudent1.DateOfBirth = DateTime.Now;
            newStudent1.IsEnrolled = false;
            newStudent1.Courses.Add("Mathematics");
            newStudent1.Courses.Add("Programming");

            var newStudent2 = new Student(1, "Maria", DateTime.Now, true);

            var newStudent3 = new Student
            {
                Id = 2,
                Name = "John",
                DateOfBirth = DateTime.Now,
                IsEnrolled = false
            };

            students.Add(newStudent1);
            students.Add(newStudent2);
            students.Add(newStudent3);

            // ------------------------------------------

            var choice = "";
            while (choice != "0")
            {
                Console.WriteLine("Choose an option:");
                Console.WriteLine("1: List all students");
                Console.WriteLine("2: Add student");
                Console.WriteLine("3: Remove student");
                Console.WriteLine("0: Exit");
                Console.WriteLine("---------------------------------------------");

                choice = Console.ReadLine();

                switch (choice)
                {
                    case "1":
                        // List all students
                        foreach (var s in students)
                        {
                            Console.WriteLine(s.Id + "\t" + s.Name);
                        }

                        Console.WriteLine("Type ID for details. x for Exit");

                        string detailsChoice = Console.ReadLine();

                        switch (choice)
                        {
                            case "x":
                                break;

                            default:
                                var results = from x in students
                                              where x.Id.ToString() == detailsChoice
                                              select x;

                                var selectedStudent = results.FirstOrDefault();

                                Console.WriteLine(selectedStudent.Id);
                                Console.WriteLine(selectedStudent.Name);
                                Console.WriteLine(selectedStudent.DateOfBirth.ToShortDateString());
                                Console.WriteLine(selectedStudent.IsEnrolled.ToString());
                                foreach(string course in selectedStudent.Courses)
                                {
                                    Console.WriteLine(course);
                                }
                                break;
                        }

                        break;

                    case "2":
                        // Add student
                        break;

                    case "3":
                        // Remove student
                        break;

                    default:
                        break;
                }
            }

        }
    }
}

Student.cs

using System;
using System.Collections.Generic;

namespace LM
{
    class Student
    {
        public int Id { get; set; }
        public string Name { get; set; }
        public DateTime DateOfBirth { get; set; }
        public bool IsEnrolled { get; set; }
        public List<string> Courses { get; set; }

        public Student()
        {
            Courses = new List<string>();
        }

        public Student(int id, string name, DateTime dob, bool enrolled)
        {
            Id = id;
            Name = name;
            DateOfBirth = dob;
            IsEnrolled = enrolled;
        }

        public void Register(string course)
        {
            Courses.Add(course);
        }

        public void Drop(string course)
        {
            Courses.Remove(course);
        }
    }
}</string></string>

Let me also give some hints:
In Students.cs you should use

Courses = new List<string>();</string>

Moreover the switch statement in Program.cs also needs a default value.