In brief, it was Poincare that critiqued the notion of simultaneity and came up with the notion of synchronisation with light signals; this in fact was a critique of Newtons absolute time which was behind the conventional notion. In more detail:
Space & time can be conceived either substantially where spacetime is thought of as a container or a medium for events, or in relatively where only the events actually exist and their relations.
Leibniz advocated a relational view against the substantial view put forward by Newton, where he claimed in the Principia:
Absolute, true, and mathematical time, in and of itself and of its own nature, without reference to anything external, flows uniformly and by another name is called duration...Absolute space, of its own nature without reference to
anything external, always remains homogeneous and immovable.
It is in reference to this model of space and time that we derive the conventional notion of simultaneity: The time on the Moon flows at the same rate as it does here on Earth, and so it is a well posed question to ask whether some event occurs at the same time as another elsewhere.
Leibniz main argument against this is that shifting everything in the universe by a uniform amount in space and time appears to give a different universe, but which he argues is in fact not. This indeterminancy suggests the relational view. (This indeterminancy, or gauge freedom, is used in Modern Field Theory to connect the symmetries of space & time with the usual conservation laws).
Light and gravity posed a problem: light was conceived as a wave, and a wave requires a medium, thus the transmission of light appeared to demand that space was filled with some kind of medium; the sun acts on the earth, but action it was argued cannot act at a distance, hence it too requires a kind of medium for transmission. (In essence, both of these arguments goes back to Parmenides disavowal of the Void - the Void being where Being is not). Hence the pressing into service of Aristotles fifth element, the aether as transmission medium of light and gravity, usually known as the lumineferous aether and conceptualised mechanically.
After Maxwells discovery of his equations, it was noticed that it showed that the speed of light was a constant. This raises the question constant in reference to what? Galileo had shown that the notion of rest was not well-posed. So the aether was also pressed into service as the bearer of the reference frame that Maxwells equations seemed to require. The most detailed scheme being that developed by Fresnel & Lorentz where the aether was completely motionless (note the resonance with Newton notion of space that is immovable); in fact, Max Born observed later that it would have been natural for physicists to identify the absolute space of Newton with that of the aether - what stood in the way was conceptualising space itself as a substance with properties, this despite William Clifford writing in On the Space-Theory of Matter:
I hold in fact
(1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.
(2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.
(3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial.
(4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity.
However the result of the Michaelson-Morley experiment which should have shown the aether to be motionless - but didn't; and nor was it saved by adopting the notion of aether dragging, where the motion of matter drags the aether along saved it as shown by the Fizeau experiment. This forced Lorentz to develop a theory of length contraction, which he claimed was a real physical effect, and local time, which he thought a heuristic; however Poincare thought otherwise, and was the trigger for his investigation into the idea of simultaneity; he writes in A measure of time:
So long as we do not go outside the domain of consciousness, the notion of time is relatively clear. Not only do we distinguish without difficulty present sensation from the remembrance of past sensations or the anticipation of future sensations, but we know perfectly well what we mean when we say that of two conscious phenomena which we remember, one was anterior to the other; or that, of two foreseen conscious phenomena, one will be anterior to the other.
When we say that two conscious facts are simultaneous, we mean that they profoundly interpenetrate, so that analysis can not separate them without mutilating them.
The order in which we arrange conscious phenomena does not admit of any arbitrariness. It is imposed upon us and of it we can change nothing.
We have not a direct intuition of the equality of two intervals of time. The persons who believe they possess this intuition are dupes of an illusion. When I say, from noon to one the same time passes as from two to three, what meaning has this affirmation?
The least reflection shows that by itself it has none at all. It will only have that which I choose to give it, by a definition which will certainly possess a certain degree of arbitrariness.
which had been anticipated by Aristotle who had remarked:
“not only do we measure change by time, but time by change, because they are defined by one another”
Poincare, attending to the constancy of light everywhere (again note here the resonance with Newton conception of time which flows everywhere uniformly) showed that a synchronisation procedure based on light signals explained local time.
Poincares explanation still referred to aether, and it was Einsteins achievement to dispense with it, or in another perspective to merge the idea of the aether with that of space itself, so that time must flow at different rates at different places, where place is not taken absolutely in Newtons sense (recall there is no acceptable notion of rest), that is place itself is in motion. And its only places whose motion with respect to each other are at rest that the conventional simultaniety still holds.