Here's a really long answer, but I can perfectly explain this discrepancy as it's a weather reporting issue.
First, temperatures are always rounded to the nearest degree in Fahrenheit in the United States, but Celsius is the preferred unit worldwide. Every temperature reading matters in determining the high and low, and your airport actually does it every second but only puts out a routine observation every hour, which is known as a METAR (Meteorological Terminal Aviation Routine) weather report. You can rest assured that the high and low refer to the highest and lowest temperatures taken by the second each day. Special METAR observations will also go out in between the hourly observations if changes in the weather that would threaten aviation like falling visibilities or a thunderstorm. Weather Underground will show those special observations, but the official hourly observations are what you see on the NWS. What you have uncovered is a flaw in Weather Underground, so let's explore why that site is wrong:
Here's what your METAR observation looked like at 4:21 AM (Z time is also UTC, which is currently 4 hours ahead of EDT, so it's 0821Z) when it was 66.2°F according to Weather Underground:
KISP 040821Z 00000KT 2SM BR BKN003 OVC014 19/19 A3010 RMK AO2
The temperature is the first line before the slash in Celsius, which I have bolded. It was 19°C at that time. If we convert to Fahrenheit, we get 66.2°F, which is what Weather Underground uses. But there's a huge problem here: That 19°C is just the temperature rounded to the nearest Celsius! Remember, the worldwide standard is Celsius, so all airport observations reflect this fact and round to the nearest degree. What you should consider is that 66°F = 18.9°C and 67°F = 19.4°C. Thus, both of those temperatures in Fahrenheit round to 19°C, so it could be either one of those! Weather Underground fails to account for this possibility. You're looking at one of the special observations above as it was issued off the hour due to the fog lowering visibilities during that time. Now, let's look at the normal hourly observation back at 3:56 AM (0756Z):
KISP 040756Z 00000KT 3SM BR FEW002 OVC014 19/19 A3010 RMK AO2 SLP194 T01940194
Again, you see the same temperature in Celsius, but note the numbers I bolded. The actual temperature in Celsius without rounding is 19.4°C. The zero out in front means that the temperature is positive in Celsius. It would be a 1 if it was negative. That temperature group only shows in the hourly observations. Weather Underground sees this group and converts 19.4°C directly to 66.9°F, which rounds to 67°F. The NWS only uses that temperature group, so it will always be correct. So, what was Islip's low this morning? Every six hours (00, 06, 12, and 18 UTC), we get the high and low from the past six hours. Here's your observation at 7:56 AM (1156Z):
KISP 041156Z 00000KT 2SM BR BKN008 OVC015 20/19 A3012 RMK AO2 SLP198 T02000194 10200 20194 51005
The numbers in bold are the high and low for the past six hours. A 1 precedes the high, and a 2 precedes the low. The high was 20.0°C (again, 0 out in front as it's positive), and the low was 19.4°C. Thus, the low was 67°F. Unfortunately, Weather Underground looks at all the observations it gathered without considering the rounding issue I have explained above. Thus, it wrongly thought the low was 18.9°C, or 66°F. The NWS uses the six-hour groups above to determine the high and low, so it always gets the correct answer unless the weather station fails.
I hope this long explanation helps you, Craig! The moral of the story is simple: Look at the actual observations from the source itself. The NWS is your best source for now, but you'll eventually learn how to read METAR reports when you take your first Meteorology classes. There are tutorials online although it's quite complex at first until you understand the METAR code. Once you have a grip on how it works, here's where you can look at METAR reports online by simply typing in your airport's ID (Islip, NY is KISP) and choosing how far back in time you want to go with the drop-down box (only goes back the past 36 hours):
http://aviationweather.gov/adds/metars/