Okay, I think I'm pretty close. I have the following going on:
$cat_id = get_cat_id('library'); wp_dropdown_categories('hierarchical=1&parent=$cat_id');
However, it doesn't work with $cat_id in there. It does work when I put the category's ID number in there (which I got when I echoed $cat_id), but obviously that presents a problem when I install the site on the real server. What should I try? Thanks!