var chars = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789";
var stringChars = new char[8];
var random = new Random();

for (int i = 0; i < stringChars.Length; i++)
{
    stringChars[i] = chars[random.Next(chars.Length)];
}

var finalString = new String(stringChars);

Not as elegant as the Linq solution.

(Note: The use of the Random class makes this unsuitable for anything security related, such as creating passwords or tokens. Use the RNGCryptoServiceProvider class if you need a strong random number generator.)

UPDATED for .NET 6. RNGCryptoServiceProvider is marked as obsolete. Instead, call RandomNumberGenerator.Create(). The code in the answer has been updated accordingly.

UPDATED based on comments. The original implementation generated a-h ~1.95% of the time and the remaining characters ~1.56% of the time. The update generates all characters ~1.61% of the time.

FRAMEWORK SUPPORT - .NET Core 3 (and future platforms that support .NET Standard 2.1 or above) provides a cryptographically sound method RandomNumberGenerator.GetInt32() to generate a random integer within a desired range.

Unlike some of the alternatives presented, this one is cryptographically sound.

using System;
using System.Security.Cryptography;
using System.Text;

namespace UniqueKey
{
    public class KeyGenerator
    {
        internal static readonly char[] chars =
            "abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ1234567890".ToCharArray(); 

        public static string GetUniqueKey(int size)
        {            
            byte[] data = new byte[4*size];
            using (var crypto = RandomNumberGenerator.Create())
            {
                crypto.GetBytes(data);
            }
            StringBuilder result = new StringBuilder(size);
            for (int i = 0; i < size; i++)
            {
                var rnd = BitConverter.ToUInt32(data, i * 4);
                var idx = rnd % chars.Length;

                result.Append(chars[idx]);
            }

            return result.ToString();
        }

        public static string GetUniqueKeyOriginal_BIASED(int size)
        {
            char[] chars =
                "abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ1234567890".ToCharArray();
            byte[] data = new byte[size];
            using (RNGCryptoServiceProvider crypto = new RNGCryptoServiceProvider())
            {
                crypto.GetBytes(data);
            }
            StringBuilder result = new StringBuilder(size);
            foreach (byte b in data)
            {
                result.Append(chars[b % (chars.Length)]);
            }
            return result.ToString();
        }
    }
}

Based on a discussion of alternatives here and updated/modified based on the comments below.

Here's a small test harness that demonstrates the distribution of characters in the old and updated output. For a deep discussion of the analysis of randomness, check out random.org.

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

namespace CryptoRNGDemo
{
    class Program
    {

        const int REPETITIONS = 1000000;
        const int KEY_SIZE = 32;

        static void Main(string[] args)
        {
            Console.WriteLine("Original BIASED implementation");
            PerformTest(REPETITIONS, KEY_SIZE, KeyGenerator.GetUniqueKeyOriginal_BIASED);

            Console.WriteLine("Updated implementation");
            PerformTest(REPETITIONS, KEY_SIZE, KeyGenerator.GetUniqueKey);
            Console.ReadKey();
        }

        static void PerformTest(int repetitions, int keySize, Func<int, string> generator)
        {
            Dictionary<char, int> counts = new Dictionary<char, int>();
            foreach (var ch in UniqueKey.KeyGenerator.chars) counts.Add(ch, 0);

            for (int i = 0; i < REPETITIONS; i++)
            {
                var key = generator(KEY_SIZE); 
                foreach (var ch in key) counts[ch]++;
            }

            int totalChars = counts.Values.Sum();
            foreach (var ch in UniqueKey.KeyGenerator.chars)
            {
                Console.WriteLine($"{ch}: {(100.0 * counts[ch] / totalChars).ToString("#.000")}%");
            }
        }
    }
}

Solution 1 - largest 'range' with most flexible length

string get_unique_string(int string_length) {
    using(var rng = new RNGCryptoServiceProvider()) {
        var bit_count = (string_length * 6);
        var byte_count = ((bit_count + 7) / 8); // rounded up
        var bytes = new byte[byte_count];
        rng.GetBytes(bytes);
        return Convert.ToBase64String(bytes);
    }
}

This solution has more range than using a GUID because a GUID has a couple of fixed bits that are always the same and therefore not random, for example the 13 character in hex is always "4" - at least in a version 6 GUID.

This solution also lets you generate a string of any length.

Solution 2 - One line of code - good for up to 22 characters

Convert.ToBase64String(Guid.NewGuid().ToByteArray()).Substring(0, 8);

You can't generate strings as long as Solution 1 and the string doesn't have the same range due to fixed bits in GUID's, but in a lot of cases this will do the job.

Solution 3 - Slightly less code

Guid.NewGuid().ToString("n").Substring(0, 8);

Mostly keeping this here for historical purpose. It uses slightly less code, that though comes as the expense of having less range - because it uses hex instead of base64 it takes more characters to represent the same range compared the other solutions.

Which means more chance of collision - testing it with 100,000 iterations of 8 character strings generated one duplicate.

Here's an example that I stole from Sam Allen example at Dot Net Perls

If you only need 8 characters, then use Path.GetRandomFileName() in the System.IO namespace. Sam says using the "Path.GetRandomFileName method here is sometimes superior, because it uses RNGCryptoServiceProvider for better randomness. However, it is limited to 11 random characters."

GetRandomFileName always returns a 12 character string with a period at the 9th character. So you'll need to strip the period (since that's not random) and then take 8 characters from the string. Actually, you could just take the first 8 characters and not worry about the period.

public string Get8CharacterRandomString()
{
    string path = Path.GetRandomFileName();
    path = path.Replace(".", ""); // Remove period.
    return path.Substring(0, 8);  // Return 8 character string
}

PS: thanks Sam

The main goals of my code are:

1. The distribution of strings is almost uniform (don't care about minor deviations, as long as they're small)
2. It outputs more than a few billion strings for each argument set. Generating an 8 character string (~47 bits of entropy) is meaningless if your PRNG only generates 2 billion (31 bits of entropy) different values.
3. It's secure, since I expect people to use this for passwords or other security tokens.

The first property is achieved by taking a 64 bit value modulo the alphabet size. For small alphabets (such as the 62 characters from the question) this leads to negligible bias. The second and third property are achieved by using RNGCryptoServiceProvider instead of System.Random.

using System;
using System.Security.Cryptography;

public static string GetRandomAlphanumericString(int length)
{
    const string alphanumericCharacters =
        "ABCDEFGHIJKLMNOPQRSTUVWXYZ" +
        "abcdefghijklmnopqrstuvwxyz" +
        "0123456789";
    return GetRandomString(length, alphanumericCharacters);
}

public static string GetRandomString(int length, IEnumerable<char> characterSet)
{
    if (length < 0)
        throw new ArgumentException("length must not be negative", "length");
    if (length > int.MaxValue / 8) // 250 million chars ought to be enough for anybody
        throw new ArgumentException("length is too big", "length");
    if (characterSet == null)
        throw new ArgumentNullException("characterSet");
    var characterArray = characterSet.Distinct().ToArray();
    if (characterArray.Length == 0)
        throw new ArgumentException("characterSet must not be empty", "characterSet");

    var bytes = new byte[length * 8];
    var result = new char[length];
    using (var cryptoProvider = new RNGCryptoServiceProvider())
    {
        cryptoProvider.GetBytes(bytes);
    }
    for (int i = 0; i < length; i++)
    {
        ulong value = BitConverter.ToUInt64(bytes, i * 8);
        result[i] = characterArray[value % (uint)characterArray.Length];
    }
    return new string(result);
}

The simplest:

public static string GetRandomAlphaNumeric()
{
    return Path.GetRandomFileName().Replace(".", "").Substring(0, 8);
}

You can get better performance if you hard code the char array and rely on System.Random:

public static string GetRandomAlphaNumeric()
{
    var chars = "abcdefghijklmnopqrstuvwxyz0123456789";
    return new string(chars.Select(c => chars[random.Next(chars.Length)]).Take(8).ToArray());
}

If ever you worry the English alphabets can change sometime around and you might lose business, then you can avoid hard coding, but should perform slightly worse (comparable to Path.GetRandomFileName approach)

public static string GetRandomAlphaNumeric()
{
    var chars = 'a'.To('z').Concat('0'.To('9')).ToList();
    return new string(chars.Select(c => chars[random.Next(chars.Length)]).Take(8).ToArray());
}

public static IEnumerable<char> To(this char start, char end)
{
    if (end < start)
        throw new ArgumentOutOfRangeException("the end char should not be less than start char", innerException: null);
    return Enumerable.Range(start, end - start + 1).Select(i => (char)i);
}

The last two approaches looks better if you can make them an extension method on System.Random instance.

Just some performance comparisons of the various answers in this thread:

Methods & Setup

// what's available
public static string possibleChars = "abcdefghijklmnopqrstuvwxyz";
// optimized (?) what's available
public static char[] possibleCharsArray = possibleChars.ToCharArray();
// optimized (precalculated) count
public static int possibleCharsAvailable = possibleChars.Length;
// shared randomization thingy
public static Random random = new Random();


// 
public string LinqIsTheNewBlack(int num) {
    return new string(
    Enumerable.Repeat(possibleCharsArray, num)
              .Select(s => s[random.Next(s.Length)])
              .ToArray());
}

// 
public string ForLoop(int num) {
    var result = new char[num];
    while(num-- > 0) {
        result[num] = possibleCharsArray[random.Next(possibleCharsAvailable)];
    }
    return new string(result);
}

public string ForLoopNonOptimized(int num) {
    var result = new char[num];
    while(num-- > 0) {
        result[num] = possibleChars[random.Next(possibleChars.Length)];
    }
    return new string(result);
}

public string Repeat(int num) {
    return new string(new char[num].Select(o => possibleCharsArray[random.Next(possibleCharsAvailable)]).ToArray());
}

// 
public string GenerateRandomString(int num) {
  var rBytes = new byte[num];
  random.NextBytes(rBytes);
  var rName = new char[num];
  while(num-- > 0)
    rName[num] = possibleCharsArray[rBytes[num] % possibleCharsAvailable];
  return new string(rName);
}

//SecureFastRandom - or SolidSwiftRandom
static string GenerateRandomString(int Length) //Configurable output string length
{
    byte[] rBytes = new byte[Length]; 
    char[] rName = new char[Length];
    SolidSwiftRandom.GetNextBytesWithMax(rBytes, biasZone);
    for (var i = 0; i < Length; i++)
    {
        rName[i] = charSet[rBytes[i] % charSet.Length];
    }
    return new string(rName);
}