AT&T has posted the specific set of steps you need to take to activate an iPhone 4 if you used the eligibility of another person on your family plan. While eligibility is transferrable, an AT&T spokesperson told me, it was initially unclear how you would activate your phone. (See "AT&T Allows Eligibility Transfers in Family Plan," 9 June 2010.)
The instructions are straightforward: charge your iPhone 4 fully, call AT&T, enter the right prompt code, and talk to a representative. (You could also visit an AT&T store, but that's likely more fuss.) Whatever you do, do not attempt to activate the iPhone before getting AT&T to handle the phone number swap. If you do, then you must go into an AT&T Store to get the iPhone 4 micro-SIM reprogrammed.
I received my iPhone 4 early - it arrived mid-day on Wednesday, June 23rd - and I called AT&T at about 5:30 PM (Pacific). Within a minute, I spoke with a customer service rep who was crackerjack, even though she hadn't processed a single iPhone 4 call yet. (Neither the early phone delivery nor the great rep were special treatment: I ordered my iPhone under my wife's name, and called the general AT&T number on the page linked above for assistance.)
There were a couple of minor hiccups. To process the SIM card registration swap, you need the ICCID, the SIM's internal ID code, and the IMEI, the phone's mobile ID (distinct from the phone's own serial number). While Apple provides a host of ways to get those numbers - see this support note - I wasn't able to use any of them.
Because the iPhone wasn't activated, I couldn't use the Settings app to get the two numbers, and I didn't want to pull out the micro-SIM if I didn't have to. Instead, I found that unlocking the unactivated phone revealed both the emergency call display, and a tiny information button (an i in a circle). Tap that i, and the ICCID and IMEI are displayed. I read this to the AT&T rep.
She updated the number registration in the system, but I still didn't see the AT&T network identifier in the iPhone 4's status bar. After she confirmed that the SIM change had taken, I plugged the phone into a USB connector on the computer to which I sync with iTunes. After following the first couple of steps, the correct phone number appeared automatically in iTunes, and the iPhone 4 lit up with an AT&T signal.
The process can work, and it was relatively simple, although it took about 20 minutes total.