How can size of the root disk in Google Compute Engine be increased?

In most cases, it will be simpler and more flexible to create a second data disk of the size you want, and attach it to the instance.

To resize a Persistent Disk (including a root disk), snapshot the disk, then create a new larger disk from the snapshot.

1. create a new disk from snapshot, but increase the size when doing so
2. create a new instance, using new, embiggened disk
3. embiggen the partition to recognize the new space (https://cloud.google.com/compute/docs/disks/persistent-disks#repartitionrootpd) (NOTE: pay special attention to the starting sector, don't just blindly hit return, you can, however blindly hit return on the ending sector)
4. sudo resize2fs /dev/sda1 (note, this step is not mentioned in the google cloud docs)

This is more like a follow-up to @user1130176's answer, but if you are running CentOS 7+, you'll need to do the following for step #4 (expanding the filesystem): xfs_growfs /dev/sda1

The new disks on CentOS 7 are of type xfs. Hope this helps, it was not very clear from all the links around.

As of 31 Mar 2016, you can resize a persistent disk online without stopping or rebooting the VM, without taking snapshots, and without having to restore it to a larger disk.

The blog post announcing the feature has the details, and you can see the docs for how to do this via the console:

Resize the persistent disk in the Google Cloud Platform Console:

1. Click on Compute Engine product tab.
2. Select Disks under the "Storage" section.
3. Click on the name of the disk that you want to resize to get to the disk details page.
4. At the top of the disk details page, click "Edit".
5. In the "Size" field, enter the new size for your disk.
6. At the bottom of the disk details page, click "Save" to apply your changes to the disk.
7. After you resize the disk, you must resize the disk partitions so that the operating system can access the additional space.

Or via CLI:

gcloud compute disks resize example-disk --size 250

Then, on Debian/Ubuntu/etc. run:

$ sudo apt install -y cloud-utils         # Debian jessie
$ sudo apt install -y cloud-guest-utils   # Debian stretch, Ubuntu
$ sudo growpart /dev/sda 1
$ sudo resize2fs /dev/sda1

or, for RedHat/Fedora/CentOS/etc.:

$ sudo dnf install -y cloud-utils-growpart
$ sudo growpart /dev/sda 1
$ sudo xfs_growfs -d /                    # CentOS 6 needs `resize2fs`

Note that some operating systems will automatically resize your partition on reboot without requiring you to do any manual steps with tools such as fdisk, resize2fs or xfs_growfs, so it should be sufficient to just resize the disk and reboot the VM for changes to take effect.