What is the meaning of the idiom 'don't let the bedbugs bite?

Answer 1

It's not an idiom -- it means just what it sounds like. It's a (usually humorous!) hope that you can sleep without getting bitten by bedbugs, and so you'll have a good night's rest.

Answer 2

In the Colonial times the wooden beds had ropes (presumably from the ships that carried the settlers to the new colonies), woven as a grid used as the bed mattress foundation. They were held in place with a spool and cog through the sides of the bed. After being slept upon over time; the ropes would stretch and loosen which require the cog to be removed and the spool to be wound tight and the cog replaced ...hence "sleep tight". The mattresses were often material stuffed with the Spanish moss hanging from the live oaks and stitched shut. Only to be reopened and re-stuffed as the material matted from use. In the Spanish moss which, grows from nutrients from the air using the Live Oak trees as their host, there lives a small red insect called a mite. Barely visible to the eye, it bites and removes blood from it's host much like lice, in order to procreate. Hence, "don't let the bedbugs bite" These mites are the original bed bugs.