A rigorous definition of thermodynamic properties Thermodynamics studies matter from a macroscopic viewpoint. We describe matter using macroscopic quantities such as pressure, volume, density, mass, etc. I read that not all macroscopic quantities are properties.
What special feature should a macroscopic quantity have for it to be considered as a property?
For example, mass flow rate, entropy generation, and exergy destruction are not properties. But I don't know why exactly.
 A: 
We describe matter using macroscopic quantities such as pressure,
volume, density, mass, etc.

The quantities you mention only cover the physical (measurable) properties of a system. There are also properties based on the the laws of thermodynamics, such as internal energy and entropy, and properties mathematically derived from other properties, such as enthalpy, Gibbs free energy and Helmholtz free energy.

What special feature should a macroscopic quantity have for it to be
considered as a property?

A common feature of each system property is that its value does not depend on the process.

For example, mass flow rate, entropy generation, and exergy
destruction are not properties. But I don't know why exactly.

Mass flow rate and entropy generation depend on the process.  Exergy is a property of the combination of the system and the environment, not the system alone. see https://www.sfu.ca/~mbahrami/ENSC%20388/Notes/Entropy.pdf and
https://en.wikipedia.org/wiki/Exergy

I'm not clear with what a property's value doesn't depend on the
process means.

It means that for each equilibrium state there is a unique set of macroscopic properties that defines the system that do not vary in time and do not depend on what path (process or set of processes) is taken to arrive at that equilibrium state. In moving from one equilibrium state to another the change in the system properties will be the same regardless of the process(es) that caused the change in equilibrium state.

Consider mass and mass flow rate for instance how can I show that
mass's value doesn't depend on the process and mass flow rate's value
depends on the process? –

Regarding mass, it is a fundamental property of a system that is a measure of the amount of matter in a system.  To quote the Hyperphysics web site "Definitions of mass often seem circular because it is such a fundamental quantity that it is hard to define in terms of something else". The site further states that, except at speeds approaching the speed of light, "mass is normally considered be an unchanging property of an object" (italics emphasis mine).
No matter what thermodynamic process is applied to a system, its mass will normally be unchanged, unless mass is added to or removed from the system by its environment, as discussed below ( or due to mass-energy conversion as in nuclear fission).
Regarding mass flow rate, it is not a system property but rather a description of the transport of a system property, i.e., its mass and all other properties associated with mass (internal energy, entropy, etc.). It applies to so called "open" systems, i.e., systems whose boundaries permit mass to flow into and/or out of the system.
Unless the mass flow rate into and out of an open system are equal, the amount of mass contained in the system will be changing (increasing or decreasing) in time.  In other words, depending on mass flow rates, the system may be in disequilibrium with macroscopic properties undefined.
Hope this helps.
A: To support the answer by @Bob D, here are quotes from Obert and Young, Elements of Thermodynamics and Heat Transfer:
"A variable is a property, if and only if, its change in value between two equilibrium states is independent of the process (is single valued).  A process occurs whenever the system undergoes either a change in state or an energy transfer at a steady state. A variable is a thermostatic property, if an only if, it is a function of other thermostatic properties." This is the clearest, most concise definition I have found.
