Although there are different statements of Mach's principle, one statement could be that acceleration is meaningless unless it can be defined relative to something. The distance stars provide a fixed reference frame against which we can define inertial frames and measure acceleration and distant matter in the universe might even somehow causes inertia.
Mach's principle seems to have fallen out of fashion in modern physics, most answers to this question Is Mach's Principle Wrong? doubt Mach's principle.
So as an alternative can a body itself provide a reference frame for acceleration?
When a body is accelerated it always gets deformed. One part of it is pushed or pulled and that influences the other parts that then also get accelerated.
A simple example is three masses attached by equal springs in an equilateral triangle shape. In an inertial frame the shape is an equilateral triangle and the potential energy stored by the springs is a minimum (diagram A)
If we take one mass and pull it, one mass is accelerated first and the triangle is deformed into an isosceles triangle, with greater stored potential energy. The other two masses are then pulled along and follow the first. The system knows it's been accelerated, but not because of any knowledge of the distant stars, but because it has been deformed, (diagram B).
If the masses are dropped in a gravitational field, (diagram C) the masses are in the equilateral triangle shape, even though they are accelerating relative to the distant stars. This seems fine with General Relativity and the equivalence principle that say the 'falling frame' is the same as an inertial frame. So again in the inertial frame there is no deformation.
So can the body itself be the reference when defining inertial frames?
In Newton's bucket experiment, the liquid in the bucket takes on the curved shape when rotating relative to the distant stars - true (independent of whether the bucket rotates too, or is suddenly grabbed and stopped). However it could be said that the liquid 'knows' it's accelerating as it's been deformed when going from the stationary state to the rotating state. The acceleration varies with different radii, for cicular motion $a=-\omega^2r$ and the liquid is deformed in the sense that parts of it further from the middle experience higher acceleration.
Thus since the acceleration isn't constant throughout the body, as in a gravitational field, the body knows, not by reference to the distant stars, but due to deformation and it's internal potential energy being increased that an acceleration is occurring.
Have there been any attempts to explain the inertia of bodies along these lines?
Could it be that the resistance to motion we experience as inertia, is due to the forces needed to cause cause compressions and deformations - and mass (inertia) doesn't exist at all?
It could be argued that it doesn't matter that an extended object is deformed during an acceleration as we know that the individual components atoms, protons, electrons etc...have mass of their own.
But the argument could be repeated at a smaller scale. Could the inertia of say a proton be due to forces needed to deform it as it accelerates, the same for the electron. It would just be necessary to deny the existence of truly point particles.