The surname of Rafael Nadal's uncle and longtime coach is Nadal. His full name is Antonio “Toni” Nadal Homar, and he coached Rafael from 1990 until 2017.
Toni began working with Rafael when the future champion was a child in Mallorca. He helped develop Nadal's heavy topspin forehand, athletic movement and famously disciplined approach to practice and competition. Their family relationship made the coaching partnership unusually personal, but Toni was also a demanding technical and mental guide.
Toni was present for the majority of Rafael's rise, including his first French Open title in 2005 and many of his 16 major victories as a coach. Carlos Moyá joined Rafael's coaching team in 2016, and Toni stepped away from coaching his nephew full-time after the 2017 season. Names such as Costa, Roig and Moyá are associated with Nadal's wider team, but none is the uncle identified in the question.