What was Moshe trying to find out, and what is the significance of the fact that Hashem told him?
Why is Moshe's name repeated?
Hadar Zekenim: Anyone whose name is repeated, he merits two worlds; 1 i.e. also Noach, Avraham and Yaakov. (Terach's name is repeated, for he did Teshuvah).
Toras Kohanim (on Vayikra 1:1): It is a Lashon of love and encouragement. 2
Refer to Bereishis 6:9:151:1.
Oznayim la'Torah: Now that the time had come to redeem Yisrael, to encourage Moshe to act without delay, or not to go even one step further wearing shoes - See Oznayim la'Torah, DH 'Va'Yomer Moshe Moshe.'
In other instances when Hashem calls a person's name twice, there is a Pesik (small interruption) indicated between the two (e.g. "Avraham, Avraham" (Bereishis 22:11); "Yaakov, Yaakov" (Bereishis 46:2)). Only here do we find "Moshe Moshe" with no interruption. Why?
Maharal #1 (Gevuros Hashem Ch. 24, p. 102; based on Midrash Rabah 2:6): Whether or not a prophecy is delivered forcefully and quickly, depends on its degree of necessity and urgency. In this case, it was difficult before Hashem that the slavery continue for even one additional moment. Therefore, He addressed Moshe quickly.
Maharal #2 (ibid.): Moshe's gift of prophecy would never cease (Hefsek) his entire life. Moshe was greater than the other Nevi'im; he was fully attached to Hashem.
Maharal #3 (ibid.): Moshe was greater than the other Nevi'im; he was fit from birth for prophecy. Other Nevi'im began as non-prophets, they had to grow to reach the level of prophecy.
Maharal #4 (ibid.): A name indicates a person's essence, his Tzurah. Doubling a name indicates complete Tzurah, which transcends the physical with no inclination towards it (and not merely Tzurah stamped onto Chomer). 1
In Maharal's terminology, Chomer is raw material or potential; and Tzurah is the function-specific form that it can take to actualize that potential. To illustrate, in a knife, the metal it is made of is the Chomer, and the sharpness that allows it to cut is the Tzurah. In a person, Tzurah means an end-goal, and actualization of potential. The average person does not fully negate his materialism; therefore, their accomplishments are affected by their physicality. Moshe, in contrast, was above the physical. Maharal concludes that the other approaches also stem from this perspective.