If I replace my router per Verizon's recommendation, will my Wi-Fi name have to be changed to a new value?

I received a notice from Verizon stating that my Actiontec router needed to be replaced. It said that my old router would still run OK past its obsolescence date, but that if I chose to keep it, they would no longer provide support for it. I agree that I should do this to stay current with related updates, but my main concern is that moving to a new router will require a change to my Wifi naming. Is this a valid concern? Are there any 'gothcas' to be aware of when replacing a router?

Question by user187840 on Super User, licensed CC BY-SA. We show a short excerpt and link to the original thread as credit.