Custom translation of Month names

Add the following to your existing functions.php file:

function ra_change_translate_text( $translated_text ) {
    if ( $translated_text == 'January' ) {
        $translated_text="Jan11";
    }
    return $translated_text;
}
add_filter( 'gettext', 'ra_change_translate_text', 20 );

If you want to change multiple strings, use this function instead:

function ra_change_translate_text_multiple( $translated ) {
    $text = array(
        'January' => 'Jan11',
        'February' => 'Feb22',
        'March' => 'Mar33',
    );
    $translated = str_ireplace(  array_keys($text),  $text,  $translated );
    return $translated;
}
add_filter( 'gettext', 'ra_change_translate_text_multiple', 20 );

You can add as many strings as you want, for example:

'April' => 'Apr44',

And the new translation can also be in a foreign language. Here’s an example with Arabic:

'May' => 'مايو',

Last notes: If you use this function, keep in mind this will change the string translation everywhere in your WordPress site: front and back end. January will become Jan11 everywhere. Try it and see.

If that is acceptable to you then you’re good to go. Otherwise, if you were looking to change the strings only on the front end (for your visitors to see), then consider making the changes to your theme or WordPress website language settings (which can be different from the admin language) – as suggested in some of the comments above.

Also, for reference, I learned about this function here: https://ronangelo.com/change-or-translate-text-on-a-wordpress-theme/. It is a very good post and if you want further information on translating text or themes in WordPress I recommend reading it.

Update to the original answer:

This function can only be used with words that are translatable – meaning, those wrapped in these functions ( __(), _e() )

However, your comment and the website you linked to have given me a clearer idea of what you are trying to achieve. Simply change the locale of the PHP used to show that date to ar-LB.UTF-8. Lebanon is one of the Arabic locales that uses أذار for March, whereas other locales use مارس. The php locale will then render the date as you want it with no need for digging into translations and strings.

So, in your WordPress theme, use this code where you want to display the date instead of the code you are already using:

<?php 
   setlocale(LC_ALL, 'ar_LB.UTF-8');
   echo strftime("%e %B %Y");
?>

In the source code of the page you linked to, the date is displayed within these lines of code:

enter image description here

Depending on how you organized your theme, that bit could be in header.php or maybe even a template part. Find it and you will see that the old PHP code to echo the date will be there between the tags. Paste the snippet I gave you in its place.

Do keep in mind that if this theme is not your own unique one and you’re not using a child theme you might lose that modification if the parent theme is updated.

I hope this helps. Please let me know if you need further assistance.