What you are trying to do is to use Symbolic references (also called 'soft references'; see perlref). This is a Bad Thing and produces only package (i.e., global) variables, but for better or worse, here it is:

>perl -wMstrict -MData::Dumper -le
"my @array_names = qw(foo baz);
 for my $array_name (@array_names) {
   no strict 'refs';
   @$array_name = ( reverse split '', $array_name );
   }
 print Data::Dumper->Dump([\@::foo, \@::baz], [qw(*foo *baz)]);
"
Name "main::foo" used only once: possible typo at -e line 1.
Name "main::baz" used only once: possible typo at -e line 1.
@foo = (
         'o',
         'o',
         'f'
       );
@baz = (
         'z',
         'a',
         'b'
       );
[download]

perlref and other documentation linked below explain why symbolic/soft references are Bad. (Don't worry about the  Name "what::ever" used only once... warnings; they'll be the least of your problems.)

A far better approach is to use a hash with 'hard' references to anonymous arrays:

>perl -wMstrict -MData::Dumper -le
"my @key_names = qw(foo baz);
 my %hash;
 for my $key_name (@key_names) {
   $hash{$key_name} = [ reverse split '', $key_name ];
   }
 print Data::Dumper->Dump([$hash{$_}], [qq{hash{$_}}])
   for @key_names;
"
$hash{foo} = [
               'o',
               'o',
               'f'
             ];

$hash{baz} = [
               'z',
               'a',
               'b'
             ];
[download]

Again, see perlref, perlreftut, perldsc and perllol for extensive info.