Time to settle this debate using some good ol' fashioned data analysis. Results
I used the CMU Pronounciation Dictionary as my base dataset, and used Python 3.4 to parse it. To plot the data I used the python package pygal.
Things to consider:
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The CMU pronounciation dictionary only contains about 130,000 words, providing around 5700 words that begin with a 'G'. This is by no means representative of the entire English language, however, I believe it provides an interesting snapshot of the issue.
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Some words in the CMU dictionary have multiple entries that contain different pronouciations. However, most of these differences were in vowels or terminating syllable pronounciations. As such, in order not to favor one 'G' phoneme over the other, I decided to omit the multiple entry words from the analysis.
NOTE: Although the official creator of the .gif file type has stated that it should be pronounced with a 'j' sound ('jif') I believe that an artificial word like this one should be adequately assimilated into the language via the most common pronounciation of the root syllable. Another acceptable assimilation would be for the general public to eventually favor one pronounciation over the other. I do not believe one man should dictate how a word is pronounced in an established language spoken by billions of people.