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If during composition, an accidental note is used, is there a method/reason for using one name instead of the other?

For example if a song is in the key of A, but a Eflat/D# note is written is there a reason to use one instead of the other?

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When to use which enharmonic equivalent is always based on the context you are in the melody and harmony. In your example in the key of A there are situations where you would use D# over Eb and there are times where you would use Eb over D#.

A simple example where you would use D# over Eb would be if you were using the progression A B E where B is V/V (the dominant chord of the dominant). The B chord contains the notes B, D#, and F# so you would obviously use the D# in this case.

A simple example where you would use Eb over D# would be if your melody was chromatically going down from E to D you would use an Eb to signify you are chromatically descending.

For more information, check out this question:

Why do notes have multiple names?

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It's all about context. Chords should be spelled correctly, and if there's a choice between enharmonic chords, then we look at the chord's context. For a melody note that's outside the chord, it might be that it's implying a chord or extension of a chord, in which case that spelling should be honored.

If none of that is the case and it's just a non-chord tone out of the blue, then write whatever is less confusing. It's less visual clutter to go E -> E♭ -> D than E -> D♯ -> D♮, for example. However, it's similarly better to go E -> D♯ -> E than E -> E♭ -> E♮.

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An accidental will change a normal note. By normal, I mean a note normally found in the current key - a diatonic note. It will depend on what has happened to that note. A couple of examples. In the key of C major, the harmony goes from C to F major. A new note comes along to help. It's a G#, making the chord a C augmented to move the harmony to F. Now, the F harmony changes subtly to get back to the C major. An A note changes to produce an F minor chord. It drops by a semitone to the same note ( in 12 et.) but this time, it's an A note that gets called Ab. In the 1st example, C+, the G has been raised to G#, whereas in the 2nd., the A has been dropped to Ab, as F minor has Ab rather than G# in it.

An exception to this may be a changed note (every accidental is a changed note, even a natural) and if there's no specific technical reason, then stick with #s in a # key, and b in a b key - but generally, the chord structure will dictate what the changed note needs to be called.

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