You can use the
date command to generate a unique(ish) filename:
/usr/local/bin/python2.7 /home/scripts/script.py > /home/scripts/logs/"`date +\%d\%m\%y_\%H\%M\%S`".log 2>&1
EDIT: Make sure to escape the percentage signs (
\%). They have a special meaning in cron and the crontab will return an error.
That way, a new log file (called
<date_time>.log) will be created every time your job runs.
$ date; touch "`date +%d%m%y_%H%M%S`.log"
Tue Sep 25 04:39:25 CEST 2012
$ ls *log
Assuming your script does not run more often than once a second (which it can't if you are running a simple crontab), you should be fine.
If, as suggested in the comments, your script is run by different users, you can add the user name to the log file's name:
/usr/local/bin/python2.7 /home/scripts/script.py > /home/scripts/logs/$(echo -n `whoami`_`date +\%d\%m\%y_\%H\%M\%S`).log 2>&1