Can I power a LilyPad with an iPhone lightning port?
If the phone has an accessible battery you could conceivably use the battery terminals to power a device such as Arduino. However, I think current smartphone batteries have complex monitoring circuits so this wont work.

If it doesn't, you'd be looking at using the USB/charger port on the phone.If your phone supports USB On-The-Go, I know that it allows phones to act as USB hosts, and according to USB-OTG specifications, the host must be able to supply at least 8mA of current between 4.4V and 5.25V.

If that's enough for you it could work.