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
[download]

(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.

Actually it does work. Perhaps this is not your exact code, perhaps you just think it does not work. Perhaps you have <Name   > ie spaces after the token? If this is HTML or even if not s/<\s*name\s*>//ig may be better. Note you don't have a /g so will only do the first instance of <name> in a given string so perhaps that is your issue?

my @ary = qw[<name> <Name> <NaMe> ];
for (@ary) {
    print;
    s/<name>/japh!/i;
    print " $_\n";
}
__DATA__
<name> japh!
<Name> japh!
<NaMe> japh!
[download]

Out on a limb: another posibility is that this isn't really perl (but you're asking here anyway because of the proliferation of things promising "Perl Compatible Regexes" which really aren't) and you're using something with a regex variant where \< and \> are actually more specific versions of what would be \b in perl (specifying beginning and end of word respectively).