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I also think that your mother's name is a variant for Trina, Nick. But as a curiosity, I'll say you that in Spain (my country), trena is a slang word for prisons.

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Is she Cornish or were her family Cornish/Cornish speaking? If not, it's likely a variant of Trina as said, Trena and Treena both occur here in England.

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If she's Cornish it would be from the common element Tre- meaning 'homestead, farm' with any number of second elements from nans (valley) to an-yer (chicken), but quite possibly from the surname Treen meaning 'homestead by a cliff castle'.

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cheers

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Devon

Trena is most likely a variant spelling of Trina, originally a nickname for Catrina. Follow hyperlinks for more details.