1. If you're using Linux with Logical Volume Manager (LVM), you can create a new EBS volume and and attach it as an additional disk device, then stop the file-related processes, unmount the filesystem, add a new Physical Volume (PV) to your Volume Group (VG), extend the Logical Volume (LV), and resize the filesystem (e.g. e2resize), then re-mount the filesystem.
2. Without LVM, attach a new, larger EBS volume, create a filesystem on it, mount it somewhere (/mnt), cp your data to it, unmount it, rm your old files from the / volume, then use the old, empty directory as the mountpoint for your new filesystem.
3. If you already had a separate data volume, and you don't want to split it across multiple EBS volumes, you could unmount it, snapshot it, re-constitute it as a new, larger EBS volume, resize the filesystem, and mount it in place of the old one.
Making an AMI and a whole new instance is totally unnecessary since resizing the system volume is an indication that you need a separate data volume.