How do I ping localhost using IPv6?

Short answer ^[rcf4291]

ping6 ip6-localhost    # Or the alias you have in /etc/hosts file (See below)
ping6 0:0:0:0:0:0:0:1  # Similar to `ping 127.0.0.1` with 7 `:` instead of 4 `.`
ping6 ::1              # The used analogous of `ping 127.0.0.1`

Changes to make working ping6 localhost

If you want to set localhost as alias for both ping and ping6 and it is not already so on your machine, it is enough to write in /etc/hosts file both the lines:

127.0.0.1       localhost
# ... and below
::1             localhost ipv6-localhost ipv6-loopback

For what it concerns the alias currently used on your system you can check your hosts file, /etc/hosts ^[1] or in a different place if on a different system ^[2] .
_{You may find ip6-localhost,ip6-loopback,ipv6-localhost,ipv6-loopback or localhost itself...}

Some words more

I understand your confusion indeed for what I read from the rfc6761 about "Special-Use Domain Names" 6.3 ^[3], about the name localhost,

Users may assume that IPv4 and IPv6 address queries for localhost names will always resolve to the respective IP loopback address.

so it should be expected as default but:

• On the current updated and untouched Ubuntu 14.04.3 LTS /etc/hosts I found the following section with ip6-localhost, ip6-loopback

   # The following lines are desirable for IPv6 capable hosts
   ::1     ip6-localhost ip6-loopback
   fe00::0 ip6-localnet
   ff00::0 ip6-mcastprefix
   ff02::1 ip6-allnodes
   ff02::2 ip6-allrouters
  

• On a Suse Enterprise 10 system I found localhost, ipv6-localhost,ipv6-loopback

   # special IPv6 addresses
   ::1             localhost ipv6-localhost ipv6-loopback
  

• On an old Debian GNU/Linux 8.2 system localhost, ip6-localhost, ip6-loopback

   # The following lines are desirable for IPv6 capable hosts
   ::1     localhost ip6-localhost ip6-loopback
  

That means that, in my limited experience, you should look in your /etc/host file and modify it or use what will you find doing, for example,

ping6 ipv6-localhost   # On some systems (maybe on Suse) 
ping6 ip6-localhost    # On some systems (maybe on Debian/*buntu)

Try:

ping6 ::1

The result would look like:

# ping6 ::1
PING ::1(::1) 56 data bytes
64 bytes from ::1: icmp_seq=1 ttl=64 time=0.040 ms
(...)
64 bytes from ::1: icmp_seq=9 ttl=64 time=0.037 ms
^C
--- ::1 ping statistics ---
9 packets transmitted, 9 received, 0% packet loss, time 7998ms
rtt min/avg/max/mdev = 0.035/0.042/0.055/0.011 ms

Ubuntu 14.04.1 LTS (Trusty Tahr):

# ping ::1
ping: unknown host ::1
# ping -6 ::1
ping: invalid option -- '6'
Usage: ping [-aAbBdDfhLnOqrRUvV] [-c count] [-i interval] [-I interface]
        [-m mark] [-M pmtudisc_option] [-l preload] [-p pattern] [-Q tos]
        [-s packetsize] [-S sndbuf] [-t ttl] [-T timestamp_option]
        [-w deadline] [-W timeout] [hop1 ...] destination
# ping -V
ping utility, iputils-s20121221

(The same for ping6 -V.)

localhost is the hostname that resolves to the 127.0.0.1 address. Your /etc/hosts file should have a separate entry for ::1, likely localhost6. So try these:

ping6 ::1
ping6 localhost6