True C static local variable replacement?

There's no direct equivalent of a C static variable in Delphi.

A writeable typed constant (see user1092187's answer) is almost equivalent. It has the same scoping and instancing properties, but does not allow the one-time initialization that is possible with a C or C++ static variable. In any case it is my opinion that writeable typed constants should be shunned as a quaint historical footnote.

You can use a global variable.

var
  dx: Integer;
  dy: Integer 

function UpdatePosition(x,y: Integer): Boolean;
begin
  if (x+dx < 0) or (x+dx > 640) then
    dx := -dx;
  ...
  MoveObject(x+dx, y+dy);
  ...
end;

You have to do the one-time initialization in the initialization section:

initialization
  dx := Trunc( Power(-1, Random(2)) );
  dy := Trunc( Power(-1, Random(2)) );

Of course this make a mess of the global namespace unlike the limited scope of a C static variable. In modern Delphi you can wrap it all up in a class and use a combination of class methods, class vars, class constructors to avoid polluting the global namespace.

type
  TPosition = class
  private class var
    dx: Integer;
    dy: Integer;
  private
    class constructor Create;
  public
    class function UpdatePosition(x,y: Integer): Boolean; static;
  end;

class constructor TPosition.Create;
begin
  dx := Trunc( Power(-1, Random(2)) );
  dy := Trunc( Power(-1, Random(2)) );
end;

class function TPosition.UpdatePosition(x,y: Integer): Boolean;
begin
  // your code
end;

Enable "Writable typed constants":

{$J+}
procedure abc;
const
  II: Integer = 45;

begin
  Inc(II);
  ShowMessage(IntToStr(II));
end;
{$J-}

procedure TForm1.Button1Click(Sender: TObject);
begin
  abc;
  abc;
  abc;
end;