every thing seems ok on the attached image. It does not even show a single " char ! What do you want to illustrate ?
can't reproduce on my test wiki. Tried ", """ and even ' anywhere : in doodle name, option, and user name (that you use for task name I believe). everything just works.
Doku elenor, php 5.6.29+dfsg-0+deb8u1, freshly downloaded doodle plugin, + a bunch of other plugins.
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Oh but wait.... why are the username/taskname written in blue on your image ? They're black on a vanilla version.
Oh I see, you use a patched version. but you didn't tell... this one ?
http://strukturpunkt.de/doodle/start
that patch obviously is buggy ! You can see that user data from the form (the user name/task name) is reused without beeing sanitized (at least properly escaped in this case). Faulty line is
$retTmp = '<tr><td class="rightalign">'.'<a href="#" onclick="javascript:setFullname(\''.$user.'\',\''. $dID . '\', ' . $optarr . '); return false">'.$user.'</a>'.'</td>';
Not sure if you use that patch, and anyway, I just don't have time to download the diff, apply, patch the patch, try, test, modify again, check again etc... In case you use this version, if you have some php skills, it may be possible to fix the bug by performing some simple replacements in $user variable. something like this
$value=hsc($user);
# REMAINING JOB TO DO : ALSO REPLACE ALL ' BY \' IN $value
$retTmp = '<tr><td class="rightalign">'.'<a href="#" onclick="javascript:setFullname(\''.$value.'\',\''. $dID . '\', ' . $optarr . '); return false">'.hsc($user).'</a>'.'</td>';