Your regex does what you want. Perhaps you forgot to apply it to several occurrences of <name>, using the /g modifier.

#!/usr/bin/perl -w
use strict;

my $name = "There is a <NaMe> in this string";
$name =~ s/\<name\>/(*)/i;
print "$name\n";

$name = "There is a <name> and another <NaMe> in this string";
$name =~ s/\<name\>/(*)/i;
print "$name\n";

$name = "There is a <name> and another <NaMe> in this string";
$name =~ s/\<name\>/(*)/gi;
print "$name\n";

__END__
__OUTPUT__
There is a (*) in this string
There is a (*) and another <NaMe> in this string
There is a (*) and another (*) in this string
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(I added a (*) to your regex to show where the replacement was taking place.)

Ther first time, your regex worked fine, because <NaMe> was the first and only occurrence. The second time it failed because there were two occurrences. The third time, with the /g modifier, it matched both names.