Not getting enough sleep does more damage than just leaving you with puffy eyes. It can cause fat to accumulate around your organs... (source)
Just don't overdo it. Too much sleep is equally bad for you. Aim for 7-8 hours of uninterrupted sleep. For more info on sleep and wellness, read Extremes of Sleep Related to Increased Fat Around Organs.
The Fitbit tracks the quality of your sleep (as well as the quantity of your daily activity, i.e. calories burned, steps taken, distance traveled).
As you fall in and out of sleep, the Fitbit tracks the movements that your body makes and can tell you how long it took you to fall asleep, how many times you woke up throughout the night and the actual time you were asleep vs the time you were in bed. (source)
You can also train yourself to sleep better by reading the Power Sleep: The Revolutionary Program That Prepares Your Mind for Peak Performance by James B. Maas (my college psychology professor who happens to be the foremost expert on sleep). This book will change your sleep patterns and make you better at whatever it is you do. You can't peak if you lack sleep...especially as you age.
Don't crash at the gym. Take a day off from your workout if you're fatigued or sleep deprived. Catch your Zzz's and start fresh tomorrow. Your organs will thank you and so will your athletic wardrobe.