Also, I found the popup menu to be a tad buggy and would sometimes be shown even though there were no matches. In the suggest method I fixed it (though it may have been because of my Style for popup.), by changing checking the count:
if(lstBox.Items.Count > 0) {
lstBox.SelectedIndex = 0;
pop.VerticalOffset = TargetControl.ActualHeight;
pop.IsOpen = true;
}

If the search routine takes a while (say, searching a large database), then it can get bogged down running on every keystroke. I added a DelayMilliseconds property and if set to something greater than zero (300 – 500ms works well), then the search won’t occur until the user pauses their typing. This is certainly not my original idea, but I thought it would work well combined with your code. Here’s the modified code:

//I split up the Suggest method as follows:

private delegate void Suggest2Del();
private System.Timers.Timer delayTimer;

private int delayMilliseconds;
public int DelayMilliseconds
{
get { return delayMilliseconds; }
set { delayMilliseconds = value; }
}

private void DelayTimerElapsed(object source, System.Timers.ElapsedEventArgs e)
{
delayTimer.Stop();
Dispatcher.BeginInvoke(new Suggest2Del(this.Suggest2));
}

private void Suggest()
{
// Skip if TargetControl are not defined or if not enough characters typed
if (TargetControl == null) return;
if (TargetControl.Text.Length < MinTypedCharacters)
{
pop.IsOpen = false;
lstBox.ItemsSource = null;
return;
}
if (SearchMethod == null) throw new NullReferenceException(”SeachMethod cannot be null.”);

if (delayMilliseconds > 0)
{
delayTimer.Interval = delayMilliseconds;
delayTimer.Start();
}
else
Suggest2();
}

private void Suggest2()
{
IEnumerable res = SearchMethod(TargetControl.Text, MaxResults);

lstBox.ItemsSource = res;
if (lstBox.Items.Count > 0)
{
lstBox.SelectedIndex = 0;
}

pop.VerticalOffset = TargetControl.ActualHeight;
pop.IsOpen = true;
}

//And added the last 2 lines to the constructor

public TextBoxAutoCompleteProvider()
{
InitializeComponent();
itemsSelected = false;
delayTimer = new System.Timers.Timer();
delayTimer.Elapsed += new System.Timers.ElapsedEventHandler(DelayTimerElapsed);
}

Just one question, is it free to use?