A:  The biggest trick is that when you create the memory for the TStrings property, it must be created as a TStringList even though it must be a TStrings in the class itself.  Here is an example of what I mean:

unit TestEdit;

interface

uses
  SysUtils, WinTypes, WinProcs, Messages, Classes, Graphics, Controls,
  Forms, Dialogs, StdCtrls;

type
  TTestEditorClass = class(TListBox)
  private
    { Private declarations }
    FStringListValues: TStrings;
  protected
    { Protected declarations }
  public
    { Public declarations }
    constructor create(AOwner: TComponent); overRide;
    procedure SetStringListValues(Value: TStrings);
  published
    { Published declarations }
    property StringListValues: tstrings read FStringListValues write SetStringListValues;
  end;

procedure Register;

implementation

procedure Register;
begin
  RegisterComponents('Samples', [TTestEditorClass]);
end;

constructor TTestEditorClass.create(AOwner: TComponent);
begin
  inherited create(AOwner);
  FStringListValues := TStringList.create;
end;

procedure TTestEditorClass.SetStringListValues(Value: TStrings);
begin
  StringListValues.Assign(Value);
end;

end.