Turn Signal Makes No Sound 2003 Saturn ION
 

I own a 2003 Saturn ION. The turn signal works but makes no sound. Does anyone know why there is no sound and how to fix it? Thanks

The clicking sound in your vehicle is controlled by the body control module, BCM. Used to be made by the flasher relay itself, but that no longer exists in your vehicle.

Many people seem to experience issues with the bulb sockets overheating causing the sockets to fail and the bulbs to fail. You have not reported any issue with functionality. Forsake of complaintness, if you did have a turn signal blinking faster than normal, that would mean that in all likelihood one of the bulbs was out. However The '04 sedans have a known issue with some of the bulb sockets which are covered in a recall.

Getting back to your stated problem, I've read everything from a defective instrument panel cluster to a defective BCM.
You don't want to swap the ion instrument panel cluster as it is Vin specific to your vehicle and will only function properly if the VIN matched BCM is installed as well. After that the mileage will be incorrect anyway.

A new BCM is prohibitively expensive for a car of this age, and if this is the only functional dysfunctional issue, I would leave it as is. As long as the turn signal bulbs light properly, driver safety is not impacted. Annoyance factor is severely impacted, but life is just sometimes that way.
@DropDead may be able to elaborate.

The chime/click 'speaker' is contained in the instrument panel cluster itself. It is controlled by a logic circuit in the instrument panel that receives inputs from the BCM logic circuit through the class 2 serial bus, and from the turn signal portion of the exterior lights system. If the rest of your chimes are working (door ajar, headlights on, warnings, etc) the BCM inputs are working. If the only thing not working is the clicking for the turn signals it's probably that logic circuit on the instrument panel itself. Replacing the instrument panel may correct this, but as Derf points out you will lose your odometer reading and it may require programming to the BCM.

Turn signal chime repair
 

My 2006 Saturn ION 3 Coupe experienced a bad chime module (no chimes at all). If you are handy with tools, a soldering iron, and plenty of patience, this can be repaired fairly inexpensively.
The chime module is embedded in the circuit board of the instrument cluster. To be certain it was just the chime and not the BCM, I pulled an instrument panel from an ION (any ION) at the junk yard, did a preliminary swap (just plugged it in) to my car and tested the chimes. All chimes worked as before EXCEPT the odometer will not budge, as was mentioned in a prior post. Now that I know I have a bad chime I needed to swap them out. This is where the patience comes in as dissembling the cluster is fairly easy but the ribbon cable to the LCD panel is very delicate. Mark where the needle pointers for the tach and speedometer are pointing before carefully prying them up and off. Once the panel is apart, unsolder the round chime module. Do the exact same to the other and swap and solder in the donar module to your board. Carefully button things up and losely install back on the car to test it out. Next, get a passenger who can monitor speed using GPS on their phone while you drive to see if you put the spedo needle back in the right spot. It may take a few attempts to get it right. As for the tach, if you have an ODB2 code reader that can read RPM's live, then do a compare (no need to drive it - you can do this in your driveway) to line this up. Like I said, it can be done, but take your time. Mine has been working perfectly now for 3+ years and thousands of miles, all for the cost of the donar IC, about $30.

I love the perseverance.