my $string = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
my ($random) = $string =~ m/(??{'.'x rand length $string })(.)/s;
print $random, "\n";
[download]

------------
UPDATE1: And here it is without the dependancy on knowing the name of the scalar holding the string (Thanks perlre).

my $string = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
my( $random ) = $string =~ m/^(??{ '.' x rand length $_ })(.)/s;
print "$random\n";
[download]

I knew there had to be a way. ;)

------------
UPDATE2: And here it is with a quantifier instead of a huge string of '.' metacharacters. The code is longer, but it's more friendly to real long target strings.

my $string = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
my ($random) = $string =~ m/^
                            (??{'.{' . int(rand length $_) . '}' })
                            (.)
                           /sx;
print "$random\n";
[download]

Enjoy!

#!/usr/bin/perl
use strict;
use warnings;

package Random_Char;
use PadWalker 'peek_my';
sub TIESCALAR { bless {}, $_[0] }
sub STORE { return 1 }
sub FETCH {my @c=keys%{{map{$_=>undef}split//,${peek_my(1)->{'$string'
+}}}};return $c[rand @c];}

package main;
tie my $rand_char, "Random_Char";
my $string = 'One bright day in the middle of the night';
my ($char) = $string =~ /($rand_char)/;
print "$char\n";
[download]

Update:
Here is another that is a bit more fragile that doesn't use modules.
It works on the form ($char) =~ "some string" =~ /($rand_char)/;

#!/usr/bin/perl
use strict;
use warnings;

package Random_Char;
sub TIESCALAR { bless {}, $_[0] }
sub STORE { return 1 }
sub FETCH {open(I,$0);my $l;while(<I>){$l=$_ and last if$.==(caller)[2
+]}
    $l =~ s/^.*=\s*(["'])(.*)\1\s*=~.*$/$2/;my @c=keys%{{
    map{$_=>undef}split//,$l}};return $c[rand @c];
}

package main;
tie my $rand_char, "Random_Char";
my ($char) = 'One bright day in the middle of the night' =~ /($rand_ch
+ar)/;
print "$char\n";
[download]