This paper contains some information about the area of filled Julia sets, though not a formula:
Yang, Guoxiao, Some geometric properties of Julia sets and filled-in Julia sets of polynomials. Complex Var. Theory Appl. 47 (2002), no. 5, 383–391. MR1906990 (2003c:37067)
There is also this more promising paper by the same author, but it is in Chinese and I can't get a copy anyway:
Yang, Guo Xiao, The area and diameter of filled-in Julia sets and Mandelbrot sets. Acta Math. Sinica 38 (1995), no. 5, 607–613. MR1372560 (96m:30040)
If someone knows these papers, I'd be grateful for any insights.
Problem A-1 in Milnor's Dynamics in one complex variable contains a formula for the area expressed as a series based on Gronwall's area theorem: $$ \pi (1 - |a_2|^2 - 3|a_4|^2 - 5|a_6|^2 - \cdots) $$ The series is said to converge slowly. The coefficients of the series can be easily computed recursively though by solving $$ \psi(w^2) = \psi(w)^2+c $$ for $$ \def\F#1{\frac{a_{#1}}{w^{#1}}} \psi(w) = w(1 + \F2 + \F4 + \F6 + \cdots) $$
