I'm not quite sure what you mean by "arbitrary string variable" and "unknown in advance", but maybe something like:

c:\@Work\Perl\monks>perl -wMstrict -le
"chomp(my $string_to_find = <STDIN>);
 ;;
 my @array_to_search = qw(Foo whathiddenever Bar 12hidden456);
 ;;
 for my $s (@array_to_search) {
   print qq{'$s' has '$string_to_find'} if $s =~ $string_to_find;
   }
"
hidden
'whathiddenever' has 'hidden'
'12hidden456' has 'hidden'
[download]

Update: In general, strings and, better yet, regex objects created by  qr// (see Regexp Quote-Like Operators in perlop) can be interpolated into and thus used to compose other regex objects:

c:\@Work\Perl\monks>perl -wMstrict -le
"chomp(my $string_to_find = <STDIN>);
 ;;
 my @array_to_search = qw(Foo whathi$denever Bar 12hi$den456);
 ;;
 my $rx_search = qr{ (?<= \d) \Q$string_to_find\E (?= \d) }xms;
 ;;
 for my $s (@array_to_search) {
   print qq{'$s' has '$string_to_find'} if $s =~ $rx_search;
   }
"
hi$den
'12hi$den456' has 'hi$den'
[download]

This illustrates both interpolation for composition and metaquoting with  \Q ... \E of an interpolated variable. See Quote and Quote-like Operators in perlop, perlre.