<A HREF="https://arxiv.org/abs/1410.1212">New Approximations for the Area of the Mandelbrot Set</A> gives the state of the art from 2014, and a related paper from 2015 is <A HREF="https://link.springer.com/article/10.1007/s11071-015-1917-4">On a numerical approximation of the boundary structure and of the area of the Mandelbrot set</A>.    

An area of 1.5052 up to five significant digits is one estimate, an upper bound is 1.68288, a lower bound is 1.3744.

My understanding is that the difficulty in obtaining an accurate estimate is due to the uncertainty whether the fractal boundary of the set contributes to the area.