True, though I was keying off your line which merely required

if (lc($name) eq lc($string)) {
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I'd say the thrust is still valid, though, assuming you are the one constructing the hash in the first place. That is, why not, at the point you create the hash key, normalize it to lowercase with:

my %hash;
my $next_key = 'What_I_am_looking_for Irrelevant_String';
my $data_for_next_key = 'xxx';

$hash{ lc $next_key } = [ ($next_key, $data_for_next_key) ];

print
    "Value associated with '$next_key' is '",
    $hash{ lc $next_key }->[1],  # or $hash{ lc $next_key }[1]
    "'\n";
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and now a test for exists would only even have to test for a lower-case version of the string_in_question. If your hash is at all sizeable, I'd think that using a little extra RAM to store the original key would be beneficial: you could avoid iterating through the keys every time you need to do this check.

This assumes that you don't have duplicate entries because of case, i.e. it will have to be okay that $hash{ aBc } clobbers $hash{ ABC }.