Do you need to fake an anchor click? From the thickbox site:
ThickBox can be invoked from a link element, input element (typically a button), and the area element (image maps).
If that is acceptable it should be as easy as putting the thickbox class on the input itself:
<input id="thickboxButton" type="button" class="thickbox" value="Click me">
If not, I would recommend using Firebug and placing a breakpoint in the onclick method of the anchor element to see if it's only triggered on the first click.
Edit:
Okay, I had to try it for myself and for me pretty much exactly your code worked in both Chrome and Firefox:
<html>
<head>
<link rel="stylesheet" href="thickbox.css" type="text/css" media="screen" />
</head>
<body>
<script src="jquery-latest.pack.js" type="text/javascript"></script>
<script src="thickbox.js" type="text/javascript"></script>
<input onclick="$('#thickboxId').click();" type="button" value="Click me">
<a id="thickboxId" href="myScript.php" class="thickbox" title="">Link</a>
</body>
</html>
The window pop ups no matter if I click the input or the anchor element. If the above code works for you, I suggest your error lies elsewhere and that you try to isolate the problem.
Another possibly is that we are using different versions of jquery/thickbox. I am using what I got from the thickbox page - jquery 1.3.2 and thickbox 3.1.