The meaning of 透 is what Elizabeth san explained.
透 is one of the most common one for the name "Tooru".
透、徹、亨 are often used for Tooru.
(For your information, what your mother described indeed is 秀 as Elizabeth san pointed out, but its pronunciation is "shuu" or "hide" when used in people's name. Since the appearance look similar to 透, your mother mixed up the meaning of these two...)