My mom likes to use Pegasus Mail under Windows because of its "selective mail download" feature. This allows her to look at the subject lines of her emails and decide which ones to download and which ones to delete without having to load the body of the email first.

I'm trying to get her to switch to Linux. If I can find a Linux email program with this feature, we're set. Do you know of any such programs?

The mail client of Mozilla can be setup to act similar to that:
It has a "Maximum message size to download from server" option.
If you set that option to say... 1kByte, then only the headers (including sender, subject, attachment filenames) of each mail will be downloaded and some lines of the body text, too, up to the 1k size limit). The message will be marked as a fragment, and will have a hyperlink in it, which can be used to download the rest of the message with one click.
Mozilla has a buil-in spam filter, too, which you can 'teach' by clicking on a 'not spam' button on the toolbar (with the possible effect of never marking any letter of that sender as spam.)

Thanks!