Meaning
Besides plants growing, in Patois 'grow' means to raise or bring up a child — 'mi granny grow mi' means 'my grandmother raised me'. Someone 'well grow' was raised with good manners ('broughtupsy'); someone who 'grow bad' was badly brought up. Being 'grow' by a relative is a common Jamaican upbringing.
Examples
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Mi granny grow mi from mi small.
My grandmother raised me from when I was little.
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Di pickney well grow, full a manners.
The child is well brought up, full of manners.
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Him grow bad, nobody never check him.
He was badly raised, no one ever corrected him.
Origin
From 'grow'; the Patois sense of 'to raise/bring up a child' is distinctive, reflecting how children are often 'grown' by grandparents or relatives.
How it’s used
'Grow' = to raise a child; 'well grow' / 'grow bad' judge the upbringing. Compare 'broughtupsy', 'bad breed'.