My 23 year old son recently got a job with pretty good pay. He wants to move into his own apartment which I think is great. Since he has no credit history, the landlord wants a co-signer. I've read a few similar questions/answers on your sight, but I this place has a1 year lease that can be broken by giving at least 1 month notice. Can I ask to have on the contract that I not be liable for no more than 2 months rent under the circumstances of their agreement?

I believe that you can ask to have anything added to a contract that you want
Them going along with it may be another story.
Ask and even if they say that is all they would want anyway ask them to write it in or reprint it in wording that you are more comfortable with.

I will challenge this as a true fact, if it is a real lease, it can not be broken with a 1 month notice, at the end of the lease, he will have to give a one month notice prior to moving is much more likely.

If he can get out of it, with a one month notice, it is a month to month rental, not a lease.

And you can ask anything you want, but I doubt the landlord will agree, since they want a co-signer to be liable for eveything that can happen.
What if your son falls asleep with a candle burning and burns down the apartment, guess who gets sued ?

If there is a problem and there is a 1000 in damages who is going to be liable for that.

And in the end, I would want that escape in one month written in plain english that it is anytime during the lease. Since a one year lease, is just that a contract that locks both sides in for one year.