Alright, just to be clear, the long answer is:
As of the date that I am writing this message, in January 2009, a user can have a thumbnail in her archives pages on WordPress.com on a free account, if:
(1) She is using a theme that supports excerpts in archives pages (some themes only support titles in excerpt pages), AND
(2) She "hand codes" the thumbnail into the post excerpt field when she is editing the post, by including it as an html img src tag.
While I personally am an html genius with years of experience and could do this with one hand tied behind my back, my sister-in-law who I'm helping with this site has no html knowledge.
Therefore, in this case, the short answer is "no."
For what it is worth if there are others who look this up and decide that this issue will tip them over the edge to use wordpress.org's downloaded version of wordpress, rather than wordpress.com: when I create wordpress sites that are NOT on wordpress.com, I use the "Yet Another Photoblog" plugin to achieve this feat, and am very happy with that plugin.
Thanks for all the thoughtful advice and amusing banter -- I'm a big wordpress fan and its community is one of the attractions.