I purchased 2 fully paid round trip tickets for my son on Continental Air under my One Pass member number. When I asked Continental (after completed trip) for my miles credit, they told me only tickets puchased in my own name will accumulate miles credit.

Is that correct?

Yes, and as far as I know, it's standard procedure for all airlines not to give credit to someone for the flight activity of another person.

The CO-specific rule (which can be found here) says the following:

Mike

The underlying principle here is that you get miles for flying, not for buying. You bought, but he flew, so if anyone is going to get miles here, it's him. The reason is to keep companies who pay for employee's travel from getting the FF points. That would eliminate much of the loyalty-generating benefit of these programs to the airline.

If your son is a Jr., or if the two of you at least have the same initial, you could in theory exploit that so one account gets credit for flights both of you take - as long as you're not both on the same plane, or on different flights with the same airline at the same time. (The ethics and practicalities of this have been discussed at length elsewhere.)

Hopefully you bought the tickets with a credit card that gives you miles or points for buying.

In any case, sign him up for CO's FF program. Some give retroactive credit for flights taken within a certain period before joining.