... I'm dealing with large hash structures and big input data ... I'm usually don't even want to supress the error (this usually means the input data is wrong and I need to deal with it) ...

The point raised by haukex here is critical: you're really dealing with input data validation, not a way to flag the use of an undef value that might be present if your input data was invalid. In other words, don't bother trying to lock the barn door after the horse has bolted. The ideal time to deal with bad data is when you get it, not after it's drifted downstream and had a chance to gum up other processes.

The only way I can think of to validate a data structure is to write a validation function specific to that structure and invoke it as soon as possible after the structure is captured:
    my %bigdata = capture_big_input_data($inputstream);
    validate_bigdata(\%bigdata);
Obviously, the bigger your data, the bigger your validation job, but that's life. (You may even decide that certain missing data fields can, in fact, be fixed up with a zero, an empty string or whatever; if so, the  validate_bigdata() function is the ideal place to do this patching.)

The other variation on this theme is when you read your input data on a line-by-line or block-by-block basis. In this case, you must figure out a way to validate each line or block as it is read — at least, that's what I would do. Anyway, you get the idea...

Thank you for the advice, but your suggested approach is highly inefficient and not useful at all in my situation.

Also (and more important), I think you're missing the point :), I'm not looking for any type of methodology to circumvent or prevent the error from happening, I just want to know if it's possible (without writing any specific checking code) to know the name of the variable that produces the error:

Use of uninitialized value in print...

I'm usually don't even want to supress the error (this usually means the input data is wrong and I need to deal with it)

If it's undefs in a Perl data structure that you seek...

sub seek_undef {
    my ($ref,$path) = (@_,"");
    if (ref $ref eq 'HASH')
        { seek_undef($ref->{$_}, $path."{$_}") for keys %$ref }
    elsif (ref $ref eq 'ARRAY')
        { seek_undef($ref->[$_], $path."[$_]") for 0..$#$ref }
    else { defined $ref or warn "undef at $path\n" }
}

seek_undef( {
    foo => { quz=>{ abc=>"def", ghi=>undef } },
    bar => { baz=>[ "jkl", undef, { ooo=>undef } ],
        blah=>undef },  uuu => undef,
} );

__END__

undef at {foo}{quz}{ghi}
undef at {bar}{baz}[1]
undef at {bar}{baz}[2]{ooo}
undef at {bar}{blah}
undef at {uuu}
[download]

As a first step we should check if newer versions still have these problems.

use strict;
use warnings;

warn "version $]";

my %hash=(a=>undef,b=>[]);
my $h=\%hash;

my @array=({a=>undef});
my $a=\@array;


warn "hash $hash{a}";
warn "hoa $hash{b}[0]";

warn "hashref $h->{a}";
warn "hoa ref $h->{b}[0]";

warn "array $array[1] ";
warn "aoh $array[0]{a} ";

warn "array ref $a->[1] ";
warn "aoh   ref $a->[0]{a} ";
[download]

version 5.014002 at /home/lanx/pm/warn_undef.pl line 4.
Use of uninitialized value $hash{"a"} in concatenation (.) or string a
+t /home/lanx/pm/warn_undef.pl line 13.
hash  at /home/lanx/pm/warn_undef.pl line 13.
Use of uninitialized value in concatenation (.) or string at /home/lan
+x/pm/warn_undef.pl line 14.
hoa  at /home/lanx/pm/warn_undef.pl line 14.
Use of uninitialized value in concatenation (.) or string at /home/lan
+x/pm/warn_undef.pl line 16.
hashref  at /home/lanx/pm/warn_undef.pl line 16.
Use of uninitialized value in concatenation (.) or string at /home/lan
+x/pm/warn_undef.pl line 17.
hoa ref  at /home/lanx/pm/warn_undef.pl line 17.
Use of uninitialized value $array[1] in concatenation (.) or string at
+ /home/lanx/pm/warn_undef.pl line 19.
array   at /home/lanx/pm/warn_undef.pl line 19.
Use of uninitialized value in concatenation (.) or string at /home/lan
+x/pm/warn_undef.pl line 20.
aoh   at /home/lanx/pm/warn_undef.pl line 20.
Use of uninitialized value in concatenation (.) or string at /home/lan
+x/pm/warn_undef.pl line 22.
array ref   at /home/lanx/pm/warn_undef.pl line 22.
Use of uninitialized value in concatenation (.) or string at /home/lan
+x/pm/warn_undef.pl line 23.
aoh   ref   at /home/lanx/pm/warn_undef.pl line 23.
[download]

tested in 5.22 with same results

 
version 5.022000 at pm/warn_undef0.pl line 4.
Use of uninitialized value $hash{"a"} in concatenation (.) or string a
+t pm/warn_undef0.pl line 13.
hash  at pm/warn_undef0.pl line 13.
Use of uninitialized value in concatenation (.) or string at pm/warn_u
+ndef0.pl line 14.
hoa  at pm/warn_undef0.pl line 14.
Use of uninitialized value in concatenation (.) or string at pm/warn_u
+ndef0.pl line 16.
hashref  at pm/warn_undef0.pl line 16.
Use of uninitialized value in concatenation (.) or string at pm/warn_u
+ndef0.pl line 17.
hoa ref  at pm/warn_undef0.pl line 17.
Use of uninitialized value $array[1] in concatenation (.) or string at
+ pm/warn_undef0.pl line 19.
array   at pm/warn_undef0.pl line 19.
Use of uninitialized value in concatenation (.) or string at pm/warn_u
+ndef0.pl line 20.
aoh   at pm/warn_undef0.pl line 20.
Use of uninitialized value in concatenation (.) or string at pm/warn_u
+ndef0.pl line 22.
array ref   at pm/warn_undef0.pl line 22.
Use of uninitialized value in concatenation (.) or string at pm/warn_u
+ndef0.pl line 23.
aoh   ref   at pm/warn_undef0.pl line 23.
[download]

Cheers Rolf
_{(addicted to the Perl Programming Language and ☆☆☆☆ :)
Je suis Charlie!}

"...it's inefficient to initialize everything..."

"Everything has it's price" (Thomas Mann)