My guess is it's because "Lily" isn't a natural Spanish spelling. If someone in Mexico heard the word "Lily" they would intuitively spell it as "Lili," while a native English speaker would switch it to "Lily".
I just looked up the Wikipedia page on Lili Ledi and even it has the word spelled differently on the same page:
"Lili Ledy contaba con dos líneas de juguetes: Lili, dedicada a las niñas, y Ledy, para los niños Una de sus muñecas más famosas fue Lagrimitas Lili cuyo tema musical de promoción tuvo una gran pregnancia y aún es recordado.[cita requerida] "Llora y llora y mueve sus manitas, sólo se contenta llevándola a pasear, a comer, a bañarse, a dormir, es lagrimitas Lilly, en su estuche cuatro en uno y cada uno para tí."
which says (rough translation);
"Lili Ledy consisted of two lines of toys: Lili, dedicated to girls, and Ledy, for boys. One of their most famous dolls was Little Tears Lili, whose theme song for promotion had great homogeneity and is still remembered. "Cries and cries and moves her hands, only happy being taken for a walk, eating, bathing, sleeping, is Little Tears Lily in this case four in one and each one for you."
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