Hello. My name is Marc. I'm a licensed attorney and I will be happy to answer your question.
Once the lease expired, it turned into a month-to-month lease. As such, the terms of the original lease - including the rent - applied during the month-to-month tenancy.
So yes - your son was bound by the terms of the original lease in general. I don't know what the terms of the lease are, so I'll assume that there was a provision that required your son to give 30, 60 or maybe 90 days notice before terminating the lease. In Kentucky, tenants are required to give a minimum of 30 days notice before terminating a month-to-month lease. Therefore, even if the original lease required a longer period of time, your son only had to give 30 days notice.
I mention all this about notice because I am assuming that the landlord is seeking 2 months rent based on a longer notice period stated in the lease. He has no right to do so. He is only entitled to one month rent if your son gave less than 30 days notice. Also, he is absolutely not entitled to the extra $100/month. Remember - the essential terms of the original lease - including the rent - was still effective. Since the new lease was not ultimately executed, there is nothing in writing stating any rent increase.
Finally - the fact that your son said "yes" on the phone and the landlord delivered the new lease to him is irrelevant. The fact is that your son did not sign the new lease. As such, it isn't worth the paper it's printed on. Leases and other agreements MUST be in writing and MUST be signed.
I hope this answers your question and provides useful information with which you can better understand your issues and options. If so, please feel free to rate my answer, as that is the only way I can get paid. And best of luck to you and your son.