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Was Hooke's law first coming from experiment or from math derivation of which Newtonian mechanical laws are the only prerequisite? Also can the law itself be reinvented in this way, or is it impossible to derive it mathematically from Newton's fundamental laws?

I want to know if Hooke's law is an axiom or a theorem, even though it was called a law...

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    $\begingroup$ The second part of the question is really about physics, not the history of physics. Once you have Newton's laws and calculus available (which, as Conifold points out, was later), Hooke's law is derivable under the assumption that the force F is a well-behaved function of the position x. Without loss of generality, suppose we define the coordinate x such that the equilibrium position is x=0. Then we have F(0)=0. If F is analytic at 0, then near 0 it converges to its Taylor series, so F(x)=-kx+..., where the minus sign is if k>0 and the equilibrium is stable. $\endgroup$ – Ben Crowell Aug 23 '15 at 14:28
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Newtonian mechanics was not yet in place when Hooke published his De Potentia Restitutiva (On Restoring Force) in 1678, Newton's Principia only came out in 1687. Hooke inferred the law from experiments not only with springs but also with wood and a "body of air", concluding:

"The power of any spring is in the same proportion with the tension thereof: that is, if one power stretch or bend it one space, two will bend it two, and three will bend it three, and so forward. Now as the theory is very short, so the way of trying it is very easie… it is very evident that… in every springing body… the force or power thereof to restore itself to its natural position is always proportionate to the distance or space it is removed therefrom".

This being said, Hooke did attempt a theoretical justification by imagining a mechanical model of springs as composed of tiny particles vibrating around some fixed positions, colliding with their neighbors and surrounded by "subtle matter". When the body is compressed the rate of collisions increases, accounting for the restoring force. When the body is stretched the force is provided by the pressure from "subtle matter". Although Hooke claims that the model leads to his law, modern analysis of his reasoning shows that it produces inverse square rather than proportional relationship. Today Hooke's law can be derived from more fundamental principles, but that requires assumptions about the molecular composition of matter and the nature of intermolecular forces, in addition to Newton's fundamental laws.

See Moyer's Robert Hooke's Ambiguous Presentation of "Hooke's Law".

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There is no derivation of Hooke's Law from Newtonian mechanics. By Hooke or anyone else. At the time of Hooke, this was an experimental law. If Hooke had some theoretical explanation, it was incorrect. The derivation of this law from other more fundamental laws is a matter of Solid State physics, and I do not think this is achieved even today.

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