We know that each elementary particle/entity must have mass, charge and spin defined. Are there any other attributes (independent) that must be defined for every elementary particle? is there any predicted attribute that can't be measured with current technology .
What property must be assigned to a fundamental particle?
That it can't be broken down into other particles.
What properties can be assigned to a fundamental particle?
Lots. Electric Charge (photon = 0), mass (gluon = 0), spin (Higgs = 0), Lepton and Baryon number, Parity, Isospin, etc.
What does fundamental mean? The root comes from "fundament":
Middle English (also denoting the base of a building, or the founding of a building or institution): from Old French fondement, from Latin fundamentum, from fundare ‘to found’.
Buildings, in order to be sound, were standing on the fundaments, large stone structures within the ground that helped the building to withstand floods and earthquakes.
The word "fundamental particle" says it all for particle physics theories. They are particles on which the specific theory is built, that models and predicts behavior of data , and is validated if correct or discarded if wrong.
So to answer such a question one must specify the theory. For the standard model particle theory, which has the symmetries of SU(3)xSU(2)xU(1), its fundamental particles are in the table . In this table the specific fundamental particles must obey the group structure above. The group structure does not depend on the masses, although it was first discovered by various mass arrangements. The group structure depends on the quantum numbers assigned to the particles in the table, including the charge. They are axiomatically defined as fundamental, and they keep being so whether before symmetry breaking or after . The theory fits the data up to now, and the theory is built based on the table and its specific attributes.
If new theories develop with new group symmetries , GUT for example, more fundamental particles would be needed. In string theories, which embed the standard model, the string itself is the fundamental entity , and the elementary particles of the table are string excitations.