You're not missing anything. Renaming stories in the Assignments panel is for convenience, ID would never rename the actual filenames on the hard drive to match, because that might cause many other problems (broken links). But there's a workaround.
The autonaming of story files when you make an InCopy assignment is an efficiency feature. The only way to control what it names those files is to not use the autonaming feature, and name the files manually the first time they're exported.
To do that, create your assignments without any stories (choose New Assignment from the Assignment panel and fill out the dialog box fields). Then export stories to InCopy format as a separate step. Start by selecting the stories that should belong to one of the assignments with the selection tool and then choose Edit > InCopy > Export Selection (any of the Export commands that don't include the word “Assignment” will work), or drag/drop the selection onto the Unassigned InCopy Content category in the Assignment panel.
You'll get a dialog box asking for the name of the InCopy files and where you want to save them. The name you enter will be used as a filename prefix if you're exporting more than one story at a time. For example, if you had selected 3 stories and entered the name “test” then the stories might be named “test-Joe Shcmoe was.icml” “test-Story1.icml” “test-As I drove to.icml” and so on. The first few words of each story are tacked on to make a unique filename.
(For the ULTIMATE control in filenaming, export each story individually. The name you enter in the Export dialog box will be the final filename. But most users export many stories at once and use creative file prefixing if that's an issue.)
In the Export dialog box, be sure and save those stories into the same project assignment folder > content folder that ID would have if it exported them automatically. It's not required but it helps keep things organized. If you don't have a content folder yet you can create it manually and put them in there.
Now your stories appear in the Unassigned category in the Assignments panel. From here you can select them and drag/drop them onto the correct assignment name. Update the assignments and you're good to go.
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