Example migrated from Codex:

The following performs a string replace on the content of the Text widget.

add_filter('widget_text', 'wpdocs_text_replace');

function wpdocs_text_replace($text, $instance, $that) {
    $search = 'welcome admin';
    $replace = 'welcome adam';
    $text = str_replace($search, $replace, $that);

    return $text;
}