Can your mother who has no will and five children choose to have her second oldest child as her executor to distribute her estate?

Without a will, she gets no say in the matter. She should write a will to specify who she wishes to be responsible for the distribution. If she doesn't have a will, the court will appoint an executor. If the heirs don't agree on who that should be, they will appoint a bank or lawyer to serve in the position, getting paid from the estate for their services.