I stopped into an Apple Store and asked a floor employee (I didn't have a Genius Bar appointment) about possible remedies. The fellow I spoke with introduced me to iOS 5's AssistiveTouch, of which I was only dimly aware, if at all (It is documented in the iPod touch User Guide For iOS 5.0 Software, and probably elsewhere as well). It can be set in:
Settings > General > Accessibility > Physical & Motor > Assistive Touch
AssistiveTouch wasn't intended as a fix for a recalcitrant Home button, but it works pretty well for that. When it's active, the AssistiveTouch button appears in one of 8 positions for the pre-5th generation iPod Touch: 4 along each of the left or right edges of the screen. The longer screen of the 5th generation iPod touch may add 2 more positions. Drag the AssistiveTouch button (even part of the distance) to change its position; it will jump to the new position.
It will likely be helpful to have a usual position for the AssistiveTouch button, so your finger automatically knows where to go. For portrait mode, which I use far more often than landscape mode, the 4 positions in the lower half of the screen are a poor choice, since they can interfere with keyboard input. When using the iPod with one hand, the 2nd position down on the left was a comfortable position for my left thumb. You might, as I did, rearrange apps and app groups on your various Home Screens to work better with the AssistiveTouch button's normal positioning. You can still open an app which is obscured by the AssistiveTouch button, but you will have to be more precise with your tap.
The concentric circles in the AssistiveTouch button are opaque white when it's first activated. After a short time, those circles and their square background become translucent, allowing easier identification of the area of the screen behind. The AssistiveTouch button is thus normally always present on the screen, either opaque or translucent -- you might want to turn it off before, e.g., watching a video. Screenshots are regrettably heavily obscured by AssistiveTouch, although some options supposedly exist for getting around that obscurement if you jailbreak your device.
Tap the AssistiveTouch button to bring up the options, one of which is a software version of the Home button. Tap that once or twice as desired, just like you would the hardware Home button.
Depending on usage, I sometimes drag the AssistiveTouch button elsewhere (because I need access to the area of the screen where it had been sitting), but much of the time it's fine on top of the position of the app (or app group) sitting in the second row and first column of a Home Screen. When drafting this revision in Notespark, it was useful to move the AssistiveTouch button to the upper left corner, the lone position out of the 8 possible which would not interfere at all with my text editing.
The software Home button is additionally quieter than the hardware Home button. It's also more gentle/discreet to, from the AssistiveTouch options, tap Device > Shake, instead of physically shaking the device (e.g., to Undo Typing).