Diagnosis
Homelink and the Genie's Intellicode rolling code system aren't syncing up. This is usually a timing issue in the programming steps.
How to Fix It
You need to program Homelink from scratch, and the order matters. 1) In your car, clear the old Homelink memory for that button (usually by holding two outside buttons for 20 seconds until the light blinks). 2) Now, at the Genie motor unit, press and release the small, square purple "Learn" button. The red LED will light up solid. 3) You have 30 seconds. Go to your car. Press and hold the Homelink button you want to program. Hold it until the car's light changes from a slow blink to a rapid blink (this can take up to 30 seconds), then release. 4) Immediately go back and press the Genie's purple Learn button again to lock in the code. Test it from the car.
If That Didn't Work
Some older Homelink systems need a "boost" from a handheld remote. If the above fails, try holding your original Genie handheld remote 1-3 inches from the Homelink button in the car while you do the programming steps. This helps transfer the correct signal frequency.
Heads Up
Don't rush step 3. You have to hold the car button long enough for it to cycle through its search mode and find the opener's signal. If you release too soon, it won't take.