The best replacement for sweepstakes is contest.

The best alternatives to sweepstakes are rated as following:
★★★★★  contest
★★★★  giveaways
★★★★  contests
★★  raffle
★★  giveaway
★★  lottery
★  promotions
★  gambling
★  prizes
  raffles
  sweepstake

Your alternative for sweepstakes is not listed? Suggest!
Speaking about sweepstakes, one probably shouldn't forget about no purchase necessary.

Other search terms related to sweepstakes and their ratings are:
★★★★★ no purchase necessary
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Google Trends

Scale is based on the average worldwide traffic of sweepstakes in all years. Click on graph for more.
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contest
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