How do I make display a specific post based on what the day of the week it is?
To get the current date and time in Twig use the string "now" and filter it with the date() filter to return it in the format you need (see the list for supported parameters).

Then do the if / elseif conditionals similar to how you'd do it in PHP. Another option would be to use the Craft specific switch tag.

{% switch "now" | date("l") %}

    {% case "Monday" %}
        {# Do Monday stuff #}

    {% case "Tuesday" %}
        {# Do Tuesday stuff #}

    {% case "Wednesday" %}
        {# Do Wednesday stuff #}

    {# etc. #}

{% endswitch %}