Reward precincts for high turnout by giving them novel, dramatic ways of casting their votes in the next election.
Honor, say, the top 5% of precincts by turnout in each state. Perhaps give special consideration for most improved, or most new registrations per capita. Fund the affairs with a progressive
tax on campaign spending.
Work with the citizens and leaders in the district to design something that uniquely captures their personality, their culture, the things that bring them glee.
In Honolulu, imagine a set of pools, each labeled with a candidate's name or a position on a measure. Give each voter a set of coconuts, one for each ballot question. Let the voters step up to a big coconut launcher, point the thing at the pools of their choice, and lob their coconuts into the water. Screen the pools off with a big wall of bamboo for privacy. Play hula music and serve donuts.
In Hershey, PA, maybe they'd give each voter a sugar-addled child, and let them lower the child face-first into a particular pool of melted chocolate that represents their position on each question. Record the results by analyzing the child's tongue for microscopic beads imprinted with identifying bar codes from each batch of chocolate.
In Texas, picture each voter giving their instructions to a monster truck driver, who would then drive over a car that corresponded to the desired vote. Each car could be equipped with a crush sensor to register the vote. The correspondence of cars to votes could change for each voter, so that the voter's instructions wouldn't reveal their vote to the driver.
Some towns might go for a theme that had more universal than local appeal, such as pulling a big lever that dumped a ton of custard over an inverted, running lawnmower, while shouting "AH ha ha ha ha ha!" (without which the vote would not officially count).
In Las Vegas... never mind. I don't want to think about what Las Vegas would do.
Retain the traditional ways of voting as well; give each voter their choice of voting method. The fanciful voting should not be a burden on the voters.