There are two versions of the origin of the phrase "sleep tight". The first theory says that sleep tight simply means "sleep well". The second theory says that it is a reference to the construction of beds before the advent of the modern mattress. Ropes were woven into a sleeping platform and the ropes would sag and cause poor sleeping. Tight ropes meant a better night's sleep, so to sleep tight meant to sleep with tight ropes making up your bed.
Sleep tight is a figure of speech that is used to show one's desire of wishing the other person to sleep comfortably and soundly. Sleep tight is taken out from the rhyming catch phrase, "Sleep tight and don't let the bed bugs bite."