You are starting your script as #!/bin/sh, which has a soft link to dash (The current version of sh which conforms with the POSIX 1003.2 and 1003.2a specifications for the shell) and dash doesn't support arrays. In debian 8 onwards dash has become default shell, so if you run ls -la /bin/sh the output will be /bin/sh -> dash
However bash still remains the default login shell, only the default /bin/sh used in shell scripts has been changed. So if you run your code on terminal, it will work just fine. More information on why this switch was made in Ubuntu can be found here.
If you want to use arrays in your script then you must start your script with #!/bin/bash
So your script works perfectly if modified like this
#!/bin/bash
NAME[0]="Hello"
echo ${NAME[0]}
More information on Dash as Sh DashAsBinSh
/bin/sh, If that shell is in fact bash, calling it by that name puts it in POSIX compatibility mode, where there are no arrays. Use shebang of#!/usr/bin/env bashinstead.