Am I understanding you correctly that writing it like this:
var $img = $('<img src=\"mysite/javascript/phone.jpg\" title=\"Edit phone numbers\" />');
Instead of the way it's written in your post gives you an error?
Also when writing code like that I'd really recommend you to switch to '' instead of "". When using "" every time the $ sign pops up PHP will think that's a variable and try for a variable replacement which probably isn't what you want. Start off by delimiting the whole customScript() in '' and change that line to:
var $img = $("<img src=\'mysite/javascript/phone.jpg\' title=\'Edit phone numbers\' />");
Finally, unless you really really have to, adding code through Requirements::customScript is really not recommended. You get problems like the ones you're having now and you're mixing PHP and js code which isn't a very good idea.