score:7
The place is not known with certainty, but some googling revealed two possible candidates:
One of them is Pechito Parado, a mountain near a very small village named Quebrada Eusebio. This place was visited by a group of young people in the Ruta Quetzal BBVA, a cultural exchange program, on the anniversary of the discovery in 2013 (more info, in Spanish, in this Spanish newspaper).
The other candidate is a mountain named Urrucallala and is mentioned in some academical journals such as Ángel Rubio's La ruta de Balboa y el descubrimiento del Océano Pacífico (1965) and an article in the Hispanic American Historical Review (1967), both in snippet view so I can't read them in full. Rubio, however, mentions both mountains and says they were very close to each other.
Upvote:2
According to Wikipedia articles both in English and Spanish that spot was the top of a mountain.
Britannica mentions “a peak in Darién” around quotes.
“A peak in Darién” is also the end of a poem of John Keats, in which he (wrongly) attributes the discovery to Hernán Cortés.
So it seems the name of the mountain or peak is not recorded.