Car has sat overnight, the car will not start right away?

Every so often, after the car has sat overnight, the car will not start right away. I turn the key and all of the electronics turn on (all dash lights come on). Usually like a 3-5 second delay until it begins to crank/start. I've replaced my battery and did a tune-up (distributor cap/rotor, spark plugs) and still have the problem at times. I've taken it to pep boys and they checked all of the electrical parts and said everything checked out fine. Any idea on why this would happen? Also, the car wouldn't start at all (same symptons) until I changed the battery. The car would start instantly for awhile afterwards and now I get my current problem (3-5 second delay sometimes). Most of the time it starts w/o any problems. Thanks for your time and assistance.