Alternatively, you wait a while after writing one string.
This helps debouncing, and if you have the button still pressed, advancing is a nice way to react:
void loop()
{
static int i;
buttonState = digitalRead(buttonPin);
if (buttonState == HIGH)
{
Serial.println(myStrings[i] );
if ( ++i > 5 ) i = 0; // loop through 0 .. 5
delay (500);
}
}
Pro : less lines of code
Con: you can't scroll faster than that, and nothing else can happen while waiting. If that bothers you, delay(500); is bad, and you should work through the other advice