Hi Jeremy and welcome to LinuxQuestions!
To run a script every 61 minutes, you're forced to set a cron job that runs every minute. In fact, if the first run is at midnight, only after 61 days it will run at midnight again. In this 61-days interval it happens to run at all possible hour and minute pairs. Said that, the crontab entry must be
* * * * * /path/to/script.sh
Following the discussion in the thread linked by pixellany, inside the script you can try to set-up a check to see if the difference between the current date and a reference date of your choice is a multiple of 61 minutes. For example (using BASH)
reftime=$(date -u -d "20100515" +%s)
timenow=$(date -u -d "$(date -u +"%Y%m%d %H:%M")" +%s)
if [ $(( ($timenow - $reftime) % 3660 )) -ne 0 ]
<your code here>
In this example the main code will run every 61 minutes starting on 15-May-2010 00:00 UTC. Calculations are made in UTC to avoid weird results at the passage from/to Daylight Saving Time.