At first, I thought the key was changed just to make the guitar phrase from the Animals' version easier to play on flute*. In the Animals' version, it's:
On guitar, that C# to D transition isn't anything special, but it's a bit difficult on flute. The fingerings for C# and D don't have any keys in common. Here's part of the fingering chart from the back of my flute book:
Lowering the phrase a whole step makes it a lot easier to play on flute. Here's the notation:
(I should note that in a few instances of this phrase, Ray Thomas adds some trills, but I didn't include those in my notation.)
Here are the relevant fingerings:
Initially, I thought that facilitating that part on flute was the reason the Moody Blues lowered the key from the Animals' version, but as I continued to think about the song, I realized that there might have been an-other reason. B minor has two sharps, but A minor doesn't have any accidentals. It might be a bit arcane, but that lack of accidentals is a musical representation of the purity mentioned in the line "I'm just a soul whose intentions are good." (There are some accidentals in the song [for example, there's a D# in the bass part during the verses and - although I think it might be more implied than outright strummed - there's an E major chord, which contains a G#], but there aren't any accidentals "built-in" as it were in A minor.) That extra-musical meaning might just be a coincidence and moving the song to A minor was just for a practical purpose, but it's still a valid exegesis.
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*Because the liner notes specifically mention the Animals' version and because the two versions of the song have roughly the same structure, I'm assuming that the Moodies based their version off the Animals' version. There are some significant differences in the lyrics though.
