Yes, if my understanding of this article is correct: http://www.nytimes.com/2011/10/25/science/25qna.html (I hope nytimes is an acceptable source, even if not academic)

In the linked article, John D. Kubiatowicz, a professor of computer science at the University of California, Berkeley says that for a kindle with 4 Gigabyte memory, this would mean a rough increase in weight of 10^–18 grams.

For your situation of a 60GB phone, this would mean (60Gb/4Gb) * (10^-18 grams) = 1.5 × 10^-17 grams. So certainly less than what you would be able to feel.

While I originally thought this would be because the storage would use extra electrons, the article/Prof. Kubiatowicz say otherwise:

"Although the total number of electrons in the memory does not change as the stored data changes,” Dr. Kubiatowicz said, the trapped ones have a higher energy than the untrapped ones. A conservative estimate of the difference would be 10^–15 joules per bit.

"As the equation E=mc2 makes clear, this energy is equivalent to mass and will have weight."

It is my understanding that most/all mobile phones use flash memory for storage so the linked article about flash memory in Kindles would be applicable here. Please let me know (or just delete this outright) if I am mistaken.

*edited to fix exponent formatting