This answer mentions Franz Ernst Neumann as the inventor of the vector potential, but who invented or discovered the scalar potential?
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Lagrange (1736-1813) in 1777, followed by Laplace (1749-1827) in 1782, was the first to introduce the scalar gravitational potential.1
Lagrange's paper, Remarques générales sur lemouvement de plusieurs corps qui s'attirent mutuellement enraison inverse des carrés des distances, was read at the Academy of Berlin on 20 October 1777.2
References
1. Assis, André K. T. Relational Mechanics and Implementation of Mach’s Principle with Weber’s Gravitational Force. Montréal: Apeiron, 2014, p. 475.
2. Duhem, Pierre Maurice Marie. Leçons sur l’électricité et le magnétisme. vol. 1. Paris: Gauthier-Villars, 1891, p. 13.
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1$\begingroup$ But the word itself was coined by George Green, if I remember correctly. $\endgroup$ Commented Mar 28, 2017 at 0:14
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$\begingroup$ @AlexandreEremenko Yes, Green is the first to use the term in English, according to the OED, in "G. Green Ess. on Applic. Math. Anal. to Electr. & Magnetism 16." $\endgroup$– GeremiaCommented Mar 28, 2017 at 15:00