Can i change the DNS in /etc/resolv.conf?

I was just having problems with the default 4.2.2.1 resolver, switched over to Google DNS (8.8.8.8 and 8.8.4.4). I wonder if DigitalOcean will have their internal/co-located DNS service to speed up these queries? Otherwise GoogleDNS works well (even if it’s not the fastest according to what I’ve read elsewhere).

Yes you are right the after reboot it gets lost. I found another solution you can edit resolvconf configuration.

edit - /etc/resolvconf/resolv.conf.d/base add whatever name server you want it works.

You can modify /etc/network/interfaces to change the name servers

auto eth0
iface eth0 inet static
        address xxx.xxx.xxx.xxx
        netmask xxx.xxx.xxx.xxx
        gateway xxx.xxx.xxx.xxx
        dns-nameservers 8.8.4.4 8.8.8.8 # modify this line

auto eth0 inet6 static << if ipv6 is enabled
       similar to ipv4 config

Then just run the following command to restart the network interface to reload new configuration.

# should be in one single line, otherwise you have to reenable the eth0 in DO console
 sudo ifdown eth0; sudo ifup eth0;