The RenameFile function will change a files name.

Example:
[RenameFile(OldName, NewName);]

Type the new name you want the file to have in Edit1, then click on the Button, select a file and click on Open.
When you use Filemanager (Or whatever) you will see that the file you selected with the OpenDialog has the name that you typed into Edit1.

procedure TForm1.Button1Click(Sender: TObject);
begin
    if OpenDialog1.Execute then
     RenameFile(OpenDialog1.FileName, Edit1.Text);
end;