Try the following steps:
1. Create a new EBS volume of the desired size. Make sure that is in the same avaialability zone as the original volume
and attach it to the target instance.
2. SSH into your instance and run the following commands:
n (for New)
p (for Primary)
Accept defaults for other fdisk options
w (for Write)
3. Once the fdisk exits, you will have to create a file system on the new partition.
presuming that ext4 was used on original partition
4. Mount the new partition to a temporary mount point using the following command:
mount /dev/sdg1 /new
5. Copy the data using the following command:
cd /var cp -ax * /new/
6. Update your /etc/fstab to use the new partition for /var
/dev/sdg1 /var ext4 defaults 0 0
7. Stop the instance, detach the original EBS Volume and keep thje smaller one.