It is unclear what kind of name could be used for a PC from Cormyr. There is no such nation in the PHB. What kinds of names do the people of the nation of Cormyr use?
They would primarily use Chondathan names.
According to the Forgotten Realms wiki, regarding the Chondathans:
Chondathans made up the bulk of the population of Altumbel, Chondath, Cormyr ...
Most Cormyrean humans would therefore use Chondathan names.
D&D third edition's Races of Faerûn, whose lore would still be relevant in Forgotten Realms canon, on page 110, states that Cormyr's humans are ethnically around 85% Chondathan, 12% Tethyrian (who the 5th edition Player's Handbook says tend to take Chondathan names), 2% Vaasan, and 1% Turami. The Vaasan aren't listed in the PHB, but the dark-skinned Turami are, so a minority of Cormyreans may use Turami names, particularly if they're of Turami ethnic background.
The wiki also has a list of inhabitants of Cormyr, which serves as a good list of examples of names.