EDIT: This is for InfluxDB v1.x only. It has been reported to not work for InfluxDB v2.x.
Make a new directory where you want to put your data and set the appropriate permissions, e.g.:
mkdir /new/path/to/influxdb
sudo chown influxdb:influxdb influxdb
Edit the following three lines of your /etc/influxdb/influxdb.conf (/usr/local/etc/influxdb.conf on macOS) so that they point to your new location:
# under [meta]
dir = "/new/path/to/influxdb/meta"
# under [data]
dir = "/new/path/to/influxdb/data"
wal-dir = "/new/path/to/influxdb/wal"
Restart the InfluxDB daemon.
sudo service influxdb restart # Ubuntu/Debian
brew services restart influxdb # macOS/homebrew
Done!
In case you want to move existing data, just simply copy the existing data (location can be found at influxdb.conf; /var/lib/influxdb on Ubuntu/Debian) to your new desired location before editing influxdb.conf and make sure the new folder has the appropriate permissions/ownership.
There is some information about backups/restores on the official docs, however just plain copying worked for me.
The above was tested on InfluxDB v1.2 on macOS/Ubuntu/Raspbian.