I have no favorite icon of St. Alexei -- they all represent him.
As to favorite pictures of Alexei Nikolaevich, I like the one where he is sitting in a chair in his mother's Mauve Boudoir looking at the camera from his left. In a clear enough exposure one can see that his face is peaceful and sweet. It is 1913, and he is recovering slowly from the incident at Spala. (Part of the series including him reading a book at a table)