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Some sources say that gravitational potential was introduced by Lagrange in 1773, and others say that it was introduced by Bernoulli in 1738. I sifted through Daniel Bernoulli's Hydrodynamica (translation by Carmody and Kobus) but could not even find the mention of velocity potential, let alone a potential for Newtonian gravity. I am hoping somebody with more knowledge could clarify this for me. Thanks a lot.

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The potential is not introduced in Hydrodynamica. Here is from The Oxford Handbook of the History of Physics, p.366:

"Gravitational potential appeared implicitly in the form of the integral of force in Johann Bernoulli’s formula for the motion of a point under the action of a central force (1712). Bernoulli did not connect his result with the conservation of vis viva; this step was taken in 1738 by his son Daniel, who also explicitly gave the form of the potential in the case of Newtonian attraction (1747). After an interval of more than twenty years, Lagrange began to make extensive use of the potential, at least as a formal mathematical entity in the equations of mechanics. He first applied it to the study of astronomical perturbations (1776), then to the general motion of masspoints subjected to mutual gravitational interactions (1779), and, finally, he inserted the potential in the equations of mechanics that bear his name (1782)".

The 1712 reference is missing, and the 1738 reference is to Commentationes de immutatione et extensione principii conservationisvirium vivarum, quae pro motu corporum coelestium requiritur. Commentarii academiae scientiarum Petropolitanae 10 (1738), 116–124. Here is a link.

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