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The meaning of the name Desdemona *is* relevant in the context of the play because of her relationship with Othello, but not because it means 'gift of God.'
Although it's unintentional, Desdemona's behaviour leads to her husband's downfall and her own subsequent murder. Therefore, the 'ill-fated' meaning is important to the plot of the play, but the idea that it could mean 'gift of God' wouldn't be relevant.
Hope that made sense!