The unstructured finite-element method has been widely used in 3D time-domain electromagnetic (EM) modeling due to its flexibility for modeling rugged topography and complex underground structures. However, how to generate high-quality grids becomes the key to high-accuracy EM responses. We have developed a weighted goal-oriented adaptive finite-element method based on hybrid posterior error estimation in combination with unstructured vector finite-element method and Backward Euler scheme to create an effective mesh. By introducing a weighting factor and adjusting the relative weights of the hybrid posterior errors, the numerical accuracy and convergence rate are greatly improved. To handle the huge difference of EM responses at different time channels, we introduce another weighting factor defined by the exponential power of time to achieve a synchronous refinement of shallow and deep meshes. The numerical experiments on a homogenous half-space model show that our algorithm performs better than the traditional adaptive method both from the accuracy and convergence. Further, we also test the effectiveness of our algorithm for modeling different abnormal bodies under a topographic earth.