This is probably quite specific to my server setup, noted for future reference. If you aren't using LVM, or have multiple Logical Volumes on your Volume Groups then it might not work so smoothly...

My server is set up with a couple of Volume Groups (removed some of the less interesting info, but seems notable that both are resizable)

$ sudo vgdisplay
  --- Volume group ---
  VG Name               chipshop-os
  Format                lvm2
  VG Status             resizable
  VG Size               110.81 GiB
  PE Size               4.00 MiB
  Total PE              28367
  Alloc PE / Size       7328 / 28.62 GiB
  Free  PE / Size       21039 / 82.18 GiB

  --- Volume group ---
  VG Name               chipshop-data
  Format                lvm2
  VG Status             resizable
  VG Size               3.64 TiB
  PE Size               4.00 MiB
  Total PE              953861
  Alloc PE / Size       655360 / 2.50 TiB
  Free  PE / Size       298501 / 1.14 TiB

Each Volume group has Logical Volumes defined as follows:

$ sudo lvdisplay
  --- Logical volume ---
  LV Path                /dev/chipshop-vg/root
  LV Name                root
  VG Name                chipshop-os
  LV Size                24.73 GiB

  --- Logical volume ---
  LV Path                /dev/chipshop-vg/swap_1
  LV Name                swap_1
  VG Name                chipshop-os
  LV Size                3.89 GiB

  --- Logical volume ---
  LV Path                /dev/chipshop-data/chipshop-media
  LV Name                chipshop-media
  VG Name                chipshop-data
  LV Size                2.50 TiB

Used disk space can be easily viewed with the command

$ df -h

Filesystem                                  Size  Used Avail Use% Mounted on
udev                                        3.8G     0  3.8G   0% /dev
tmpfs                                       769M  9.1M  760M   2% /run
/dev/mapper/chipshop--vg-root                25G   14G  9.7G  59% /
tmpfs                                       3.8G   12K  3.8G   1% /dev/shm
tmpfs                                       5.0M     0  5.0M   0% /run/lock
tmpfs                                       3.8G     0  3.8G   0% /sys/fs/cgroup
/dev/sda2                                   473M  177M  273M  40% /boot
/dev/sda1                                   511M  3.4M  508M   1% /boot/efi
/dev/mapper/chipshop--data-chipshop--media  2.5T  2.3T   61G  98% /data/media
tmpfs                                       769M     0  769M   0% /run/user/1000

I can see that /data/media is limited on space, so I can extend the Logical Volume using the command

$sudo lvextend --size +75G /dev/chipshop-data/chipshop-media

Size of logical volume chipshop-data/chipshop-media changed from 2.52 TiB (661760 extents) to 2.60 TiB (680960 extents).
  Logical volume chipshop-media successfully resized.

Assuming this all goes smoothly resize2fs can be used to extend the partition to the full size of the logical volume.

$ sudo resize2fs /dev/chipshop-data/chipshop-media

resize2fs 1.42.13 (17-May-2015)
Filesystem at /dev/chipshop-data/chipshop-media is mounted on /data/media; on-line resizing required
old_desc_blocks = 162, new_desc_blocks = 167
The filesystem on /dev/chipshop-data/chipshop-media is now 697303040 (4k) blocks long.

Size can then be checked again with df -h - now showing the new size "/dev/mapper/chipshop--data-chipshop--media 2.6T 2.4T 131G 95% /data/media"