Will an iPhone only activate using the SIM card that it was originally activated with?
No - any valid SIM from the correct carrier will activate a phone that is locked to one carrier. 

In the rarer case - iPhone 3G was sold or is now authorized unlocked in which case, you don't need any SIM to activate the device after a restore operation.

If the device is carrier locked (in the US - AT&amp;T is the likely carrier), you can either buy a valid SIM from the correct carrier. A prepaid line or locating the SIM that was last used to activate the phone are good options for low expense. 

Apple has articles that can guide you in this process:


Using an iPhone without a wireless service plan - This covers setting up an old iPhone like an iPod touch.
iOS: How to back up your data and set up as a new device - The end of this covers restore, set up as new and more explanation of carrier plans.
About swapping the SIM card
iPhone: Troubleshooting authorized unlock issues