First work out which disk it is in the array (e.g. /dev/sda), then use lshw to get the serial number of the disk.
Now mark all the partitions on the disk as degraded
mdadm --manage /dev/md0 --fail /dev/sda1 mdadm --manage /dev/md0 --remove /dev/sda1
Now power down the machine and replace the failed drive using the serial number from lshw to identify it if necessary
Once the new drive is in, power the machine up, recreate the partition setup on it in the same way as the old drive was and then re-add the drive to the raid array with
mdadm --manage /dev/md0 --add /dev/sda1
You can then cat /proc/mdstat to see the drive start rebuilding.
To speed up the rebuild if the machine isn't otherwise too busy you can adjust the values of
Try setting the min to 50000 and the max to 200000
You can also turn bitmaps on to speed up the rebuild with
mdadm --grow --bitmap=internal /dev/md0
then turn them off once the rebuild is complete with
mdadm --grow --bitmap=none /dev/md0