As noted earlier, the mountain where God appeared to the Children of Israel is called both Sinai and Horeb. As discussed earlier, the Documentary Hypothesis notes that P and J exclusively use Sinai and E and D exclusively used Horeb.
I have not yet been able to find a traditional explanation of the use of these two names. Rashi does not mention anything. The Talmud notes that these names refer to the same mountain, and then notes the derivations of the names:
What is [the meaning of] Mount Sinai? The mountain whereon there descended hostility [sin'ah] toward idolaters. And thus R. Jose son of R. Hanina said: It has five names: [...] Whilst what was its [real] name? Its name was Horeb. Now they disagree with R. Abbahu, For R. Abbahu said: its name was Mount Sinai, and why was it called Mount Horeb? Because desolation [hurbah] to idolaters descended thereon.
Without getting into the merits of this claim, it simply is addressing a different question. Regardless of how the names were derived and what they mean, why is it that one name is used in certain places and another name is used in other places?
If anyone has an explanation from traditional sources, please leave a comment.