I'll invite your suggestions for forecast contests to hold. In the mean time, some results from forecast contests at my work.
The winter contest was to predict the date of the first 2 inch (5 cm) snowfall at our official weather station. It never happened. I came close to predicting the date, sort of. Since we've had some memorable storms on or near President's day (February 18th this year), I went with that. Nothing noteworthy that day this year. But the next guesser was for May 1, so when we were getting forecasts of significant snow (4-8 inches, 10-20 cm) in mid-March, I was hopeful. Only 1.7" at the official station, though, so no luck for me. (If only we'd used any of the other area stations! All beat 2".) This is our second straight year of not having even one day with 2" of snow. Should probably adjust the standards to 1" (2.5 cm).
The summer contest had a winner before entries had even closed. One of the contests was to predict the first day that the official station would exceed 90 F (32 C). Entries open to the 30th of April, it happened the 10th if I remember correctly. The 7th earliest date ever. Spring here, apparently was April 8th and 9th. We're now on summer. Note to future: have to close entries on the summer forecast contest on April 1st or earlier. (Our earliest ever 90 F day was apparently late March -- 27th, iirc).
Both contests suggest that traditional weather forecast contests need some updating for changing climate.
For here, a couple of contests that came to mind, in addition to the 'traditional' guessing of the September average Arctic sea ice extent, are to guess when the atmospheric CO2 levels for Mauna Loa monthly average will pass 400 ppm, and when it will pass 150% of pre-industrial (420 ppm). One that can be done annually, guess the first week when Arctic sea ice extent will fall below the climatological (1979-2000) minimum extent, and guess how many weeks the ice will remain below that minimum.