Meaning: Sleep well.

Origin
'Sleep tight' is a very well-used phrase in many parts of the English-speaking world. It's common at bedtime in the form of the rhyme "good night, sleep tight, don't let the bedbugs bite".

There are many meanings of the word 'tight' and, unsurprisingly, there are several theories going the rounds as to the origin of 'sleep tight'. One is that the phrase dates from the days when mattresses were supported by ropes which needed to be pulled tight to provide a well-sprung bed. This was the notion that was put forward on a 2008 BBC antiques show, when the presenter lay on an oak settle to demonstrate the support provided by the understringing and to confidently pronounce "hence the expression 'night, night, sleep tight'". This explanation seems unlikely, as it is the bed rather than its occupant that is tight and no one (in my experience) ever wishes furniture a good night's sleep. He would have had more luck had he opted to say that 'settle down to sleep' derives from 'settle' or 'seat' - which it does.

The phrase 'sleep tight' itself was common in the late 20th century, and there could hardly have been a better way of cementing any phrase into the popular consciousness than by Lennon and McCartney using it in the lyrics of a song at the height of Beatlemania. That's where it found itself, in Good Night on the White Album in 1968:

Now it's time to say good night,
Good night. Sleep tight.

From: http://www.phrases.org.uk