Find the date for the nearest day of the week, in this case Fri & Sat
I think dateOfNearestDay would be more intuitive and useful if it would return the date object, 
after all you are dealing with dates here. This would also simplify the caller code, who now works with a date object instead of an array.

Keep a separated method to format the date the way you want to and you are good to go.
Besides of this two things you wrote nice code overall.



(function() {

  var startingDate = new Date();

  function dateOfNearestDay(startingDate, nearestDay) {
    var nearestTime = new Date(startingDate.getTime());

    if (startingDate.getDay() == 6 &amp;&amp; nearestDay == 5) {
        nearestTime.setDate((startingDate.getDate() + 
           (7 + nearestDay - startingDate.getDay()) % 7) - 7);
    } 
    else {
        nearestTime.setDate(startingDate.getDate() + 
           (7 + nearestDay - startingDate.getDay()) % 7);
    }

    return nearestTime;
  }

  function getMonthAndDay(date){
      var monthNames = ['January', 'February', 'March', 'April', 
      'May', 'June', 'July', 'August', 'September', 
      'October', 'November', 'December'];

      return monthNames[date.getMonth()]+ " " + date.getDate(); 
  }
  
  var nearestDateFriday = dateOfNearestDay(startingDate, 5);
  var nearestDateSaturday = dateOfNearestDay(startingDate, 6);

  var fridaysDate = document.getElementById('fridaysDate');
  var saturdaysDate = document.getElementById('saturdaysDate');

  fridaysDate.textContent = getMonthAndDay(nearestDateFriday );
  saturdaysDate.textContent = getMonthAndDay(nearestDateSaturday );

}());
&lt;div id="fridaysDate"&gt;&lt;/div&gt;
&lt;div id="saturdaysDate"&gt;&lt;/div&gt;