It is actually from the 1990s (not older), but this sounds similar to the 1991 novel by Robert Sawyer Golden Fleece
Points that seem to match:
Involved a human colony ship
The Argo is a colonization ship on its way to Eta Cephei IV with more than ten thousand humans aboard. But this is no sleeper ship.
where a member of the crew has unexpectedly died.
The story begins as JASON (the ship's AI) murders crew member Diana Chandler and is nearly successful in making it look like suicide.
The main characters were her estranged husband and the ship's AI.
Diana's ex-husband Aaron Rossman believes that Diana has killed herself because of his actions. His affair during the final months of their marriage was not a secret as he believed. Aaron gradually sees past his grief and deduces that Diana was murdered.
The AIs were careful in the crew selection to make sure they had diverse genetic stock to reduce inbreeding concerns. The ship's AI calculates who the deceased's closest relative on board would be, then decides not to tell people because it doesn't want to draw attention to the level of separation.
The best and brightest (and youngest) of Earth have boarded the 'Argo,' a colonization ship bound for Eta Cephei IV.
The cover had a reference to HAL from 2001 space odyssey, something to the effect of "move over HAL, there is a new computer and this one is capable of murder
JASON is a bit like HAL from 2001: A Space Odyssey, but has more humans to talk with. And lie to.
Cover blurb below:
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