DO NOT USE THE SPACE IN /mnt! This is ephemeral storage and will not persist across reboots - if you put stuff in there IT WILL BE LOST. Some linux distros mount the ephemeral storage up for you as a convenience, use it for temp or swap.
You have a couple options.
Increase Root Drive Size
You can't resize /dev/sda1 on the fly. You can resize it on launch, however. This will require downtime, but what you can do is
Snapshot the existing instance into an AMI. This should give you ami-yyyyyy (I hope you are doing this or something like it already to make backups over time!)
ec2-create-image --name my-image-name-v1.0 --description "My lovely Web Server" i-xxxxxxx
Run a new instance of that image with a larger root drive size
ec2-run-instances -k ssh-key -z us-east-1b -t c1.medium -b "/dev/sda1=:50" ami-yyyyyy
Now you are running instance i-zzzzz. Depending on what Linux you are using you may need to then resize the file system to get the additional space. On Ubuntu, on the box:
sudo resize2fs /dev/sdf
Now swap in i-zzzzzz for i-xxxxxx in your elastic IP or ELB or DNS or however you're advertising it to the world.
Add A Second Drive
This is probably better - marginally more expensive, but best practice is NOT to put a bunch of stuff on your root drive, as if it fills up with logs or files you're going to crash and have a sad time of recovery.
Create an EBS volume of the desired size, let's say 20 GB. This gives you a volume, vol-yyyyyy.
ec2-create-volume -z us-east-1b -s 20
Attach the volume to your instance
ec2-attach-volume vol-yyyyyy -i i-xxxxxx -d /dev/sdf
On the instance, create a filesystem on it and mount it
sudo mkfs -t ext3 /dev/sdf
sudo mkdir -p /web
sudo mount /dev/sdf /web
Move your web root over to there.
Add the new drive permanently to /etc/mnttab
/dev/sdf /opt/apps ext3 defaults,rw 0 0
Snapshot your new image to AMI as in step 1 - always a good practice.
This also has the benefit of being able to back up that EBS separately by just snapshotting the volume, and also if you need to kill one server and bring up another, you can detach the /web EBS volume from one and attach it to the other, making data migration easy.