Why do you say sleep tight?

Answer 1

The phrase "sleep tight" originated when mattresses were set upon ropes woven through the bed frame.(before box-springs were invented) To remedy a sagging, uncomfortable matress caused by sagging and loose ropes, one would use a bed key to tighten the rope. Means have a good sleep

Answer 2

"Good night. Sleep tight. Don't let the bedbugs bite." Before the box spring, well-made bed-frames had ropes that needed to be tightened periodically against sagging, and canopies to keep creatures from falling out of the thatch roof into the bed. Perhaps sleeping "tight" means in a comfortably well-suspended bed.

This explanation may of course be "folk etymology." The expression "sleep tight" was first recorded in the early 20th Century, long after the box spring had replaced the 'rope sprung' system, but the earliest recorded use of a word or phrase and its earliest actual use are not necessarily the same thing, by a long shot.

There are a number of theories which are all inconclusive. It just means 'sleep soundly.' The word tight in the English language has been closely linked with the words 'soundly,' 'well' or 'properly.'