Zip, check the "Weather links" link in the Feature topics box above on the left. That page has links to all the related weather links we can find that apply to Mt. Whitney and the surrounding area.
sjarelkwint, the NOAA "Whitney Summit weather" link will get you a pretty close prediction, although it is for a lower elevation. Temps would be several degrees lower for the summit. And the White Mountain Summit weather station can give you actual values at White Mountain; its weather is quite similar to Whitney, only drier.
People often sleep at the summit, although altitude issues could render it a restless night. It depends on how well acclimated you are. Ear plugs would help if you sleep in the hut -- any breeze will cause the metal roof to rattle some.