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My guess is this:
The root is ἀλεκ- (as in ἀλέξω, ἀλκή ...).
If the ending is -σις, the word would refer to an activity and its result, like you said.
But since the names Ἄλεξις and Ἀλέξιος were used somewhat synonymously, the ending could be -ιος (instead of -σις), which would refer to the descent and origin of a person. So if the origin is "helping", "defending", the person might be a "defender", "helper". This is, how I'd explain the meaning given by this site.