Should you hike Half Dome if bad weather is forecast on your permit day?

No. The National Park Service instructs hikers not to attempt the cables in bad weather, and the guidance is specific to lightning and wet rock. The park states plainly: "Do not attempt the ascent if: Storm clouds are in the area" or "The ground is wet," adding that "Nearly all fatal falls from the cables have happened when the rock was wet, which make the rock (and cables) very slick." Source: https://www.nps.gov/yose/planyourvisit/halfdome.htm

The summit itself is treated as hazardous terrain in a storm. NPS warns that "The summit of Half Dome is a dangerous place during a lightning storm," and advises checking the forecast beforehand and aiming to reach the top early, since afternoon thunderstorms are common even though they "can occur at any time." If a storm moves in, the instruction is to turn back before the summit, or to leave the summit area immediately while still using caution on the cables and steps if already there. The park also notes the summit runs 15 to 20°F cooler than the valley with frequent wind, so hikers should be ready for rain and cold even without a storm. Source: https://www.nps.gov/yose/planyourvisit/halfdome.htm

Because the cables are only safe in dry conditions, NPS repeats elsewhere that "The leading cause of fatal falls from the cable route is being on the cables when the rock and/or cables are wet." This is treated as a hard stop, not a judgment call to push through.

Skipping the hike for weather does not have to mean losing the permit fee. The $10 per-person charge that applies once a day-hiker permit is awarded "is fully refundable if you cancel your permit by 11:59 pm Pacific time the day before your hiking date or if the cables are not up on the date for which your permit is valid." Cancelling ahead of a forecast storm, rather than risking the climb, is consistent with both the park's safety guidance and its refund policy. Source: https://www.nps.gov/yose/planyourvisit/hdpermits.htm

In short: if thunderstorms, rain, or wet rock are expected on a Half Dome permit date, the Half Dome cables should not be attempted that day.