He has been working on it for the better half of the decade.
What does this sentence suggest?
In this context, the better half is a somewhat non-standard variant of...
best part of something - nearly all of something, the majority, most
...so really all it means here is for more than five years. Or feasibly, for more than 15 months if we assume by the decade he means the period of ten years starting from 2012 (which I seriously doubt! :).
Usually, the (or my) better half is a slang/ironic term for one's wife (sometimes husband).