The //= operator is the assignment operator version of the // or ‘logical defined-or’ operator.
In the context of a my variable declaration, the variable is initially undefined so it is equivalent to assignment (and would be better written as my $i = '08';). In general, though,
$i //= '08';
is a shorthand for:
$i = (defined $i) ? $i : '08';
It is documented in the Perl operators (perldoc perlop) in two places (tersely under the assignment operators section, and in full in the section on ‘logical defined-or’). It was added in Perl 5.10.0.