G'day bliako,

Here's another way: enum for the order; and, an Orcish Manoeuvre for efficiency.

#!/usr/bin/env perl

use 5.010;
use strict;
use warnings;

use enum qw{sun mon tue wed thu fri sat};

say '*** Check Orcish Manoeuvre is working ***';
say _canon_day_test($_) for qw{Mon mon Monday monday} x2;

say '*** Your sort example ***';
my @weekdays = qw/Monday Saturday Thursday/;
say for sort { _canon_day($a) <=> _canon_day($b) } @weekdays;

say '*** More complex example ***';
my @many_form_weekdays = qw{
    thurs tue Thu Tues Thursday tuesday
    Fri Sun sat Fri Sun sat SATURDAY FRIDAY SUNDAY
    Wednesday Mon wed monday
    sun Sat friday Friday Sunday
};
say for sort { _canon_day($a) <=> _canon_day($b) } @many_form_weekdays
+;

sub _canon_day {
    my ($day) = @_;
    state $day_for = {};
    return $day_for->{$day} ||= eval lc substr $day, 0, 3;
}

sub _canon_day_test {
    my ($day) = @_;
    state $day_for = {};
    say "'$day' needs Orc" unless exists $day_for->{$day};
    return $day_for->{$day} ||= eval lc substr $day, 0, 3;
}
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_canon_day_test() is identical to _canon_day() except for the say statement: it's just for testing.

You may want to add some validation code, but that wasn't part of your question.

Output:

*** Check Orcish Manoeuvre is working ***
'Mon' needs Orc
1
'mon' needs Orc
1
'Monday' needs Orc
1
'monday' needs Orc
1
1
1
1
1
*** Your sort example ***
Monday
Thursday
Saturday
*** More complex example ***
Sun
Sun
SUNDAY
sun
Sunday
Mon
monday
tue
Tues
tuesday
Wednesday
wed
thurs
Thu
Thursday
Fri
Fri
FRIDAY
friday
Friday
sat
sat
SATURDAY
Sat
[download]