Trial and error mostly, involving any number of the following steps in various sequences:

Turn off BT on iPhone
Turn on BT on iPhone
Forget device on iPhone
Try to re-pair to E510 on iPhone
Turn off BT on E510
Turn on BT on E510
Forget device on E510
Try to re-pair to iPhone on E510
Quit Garmin Connect
Start Garmin Connect
Refresh Garmin Connect

It eventually works. I still get 2 E510s in Garmin Connect, no idea why, and I still occasionally have trouble uploading rides via the E510, even after things are successfully paired up. :shrug: