I have a server hosted at Mythic Beasts and I'd run out of disk space. They have instructions but I'm not experienced enough to be able to follow them as is, though I broadly followed it to make sure I was roughly on track.

Here's what I did so I can do it again. I do not necessarily think I did it in the optimal or correct way, but it did work. YMMV.

> df
...
/dev/mapper/nicecupoftea2--vg-root 156193244 135604120  12628756  92% /
...

> sudo lsblk

...
sr0                           11:0    1   667M  0 rom  
vda                          252:0    0   200G  0 disk 
├─vda1                       252:1    0   487M  0 part /boot
├─vda2                       252:2    0     1K  0 part 
└─vda5                       252:5    0 159.5G  0 part 
  ├─nicecupoftea2--vg-root   253:0    0 151.5G  0 lvm  /
  └─nicecupoftea2--vg-swap_1 253:1    0     8G  0 lvm  [SWAP]


> sudo parted /dev/vda

(parted) print free
Model: Virtio Block Device (virtblk)
Disk /dev/vda: 215GB
Sector size (logical/physical): 512B/512B
Partition Table: msdos
Disk Flags: 

Number  Start   End     Size    Type      File system  Flags
        32.3kB  1049kB  1016kB            Free Space
 1      1049kB  512MB   511MB   primary   ext2         boot
        512MB   513MB   1048kB            Free Space
 2      513MB   172GB   171GB   extended
 5      513MB   172GB   171GB   logical                lvm
        172GB   215GB   43.0GB            Free Space

(parted) mkpart 
Partition type?  primary/logical? primary                                 
File system type?  [ext2]?                                                
Start? 173GB
End? 215GB
(parted) print                                                            
Model: Virtio Block Device (virtblk)
Disk /dev/vda: 215GB
Sector size (logical/physical): 512B/512B
Partition Table: msdos
Disk Flags: 

Number  Start   End    Size    Type      File system  Flags
 1      1049kB  512MB  511MB   primary   ext2         boot
 2      513MB   172GB  171GB   extended
 5      513MB   172GB  171GB   logical                lvm
 3      173GB   215GB  41.7GB  primary   ext2         lba

> sudo fdisk -l

Device     Boot     Start       End   Sectors   Size Id Type
/dev/vda1  *         2048    999423    997376   487M 83 Linux
/dev/vda2         1001470 335542271 334540802 159.5G  5 Extended
/dev/vda3       337891328 419430399  81539072  38.9G 83 Linux
/dev/vda5         1001472 335542271 334540800 159.5G 8e Linux LVM

> sudo pvcreate /dev/vda3
Physical volume "/dev/vda3" successfully created.

> sudo vgdisplay
  --- Volume group ---
  VG Name               nicecupoftea2-vg
  System ID             
  Format                lvm2
...

> sudo vgextend nicecupoftea2-vg /dev/vda3
  Volume group "nicecupoftea2-vg" successfully extended

> sudo lvextend -l +100%FREE nicecupoftea2-vg/root
Filesystem at /dev/mapper/nicecupoftea2--vg-root is mounted on /; online resizing required
old_desc_blocks = 10, new_desc_blocks = 12
The filesystem on /dev/mapper/nicecupoftea2--vg-root is now 49911808 (4k) blocks long.

> df
Filesystem                         1K-blocks      Used Available ...
/dev/mapper/nicecupoftea2--vg-root 196321316 135604972  51129744  73% /
...

Various links

How to resize a logical volume with 5 simple LVM commands guide helped and so did Creating and managing partitions in Linux with parted.

Overview of Logical Volume Manager (LVM) was handy, and How to Increase the size of a Linux LVM by expanding the virtual machine disk (steps to take, what the hex things mean).

I didn't know whether to use ext4 or what so I went with the default/existing, ext2 (some info). I didn't know if it should be logical or physical, my conclusion was that it didn't matter for my purposes.