Some of this has emerged in the discussion while I was researching and writing it, but I’ll post it just the same.

I don’t disagree with Tony’s answer at all. “Grown” certainly does mean “adult”. I just want to offer an explanation of why this writer has used this word and how it is related to the connotations of “boy”.

A lawn boy is a servant who tends someone’s lawn. Wealthy suburban Americans, and even not-so-wealthy ones, attach enormous importance to having a large, well-tended lawn. In his recent book At Home, Bill Bryson tells us that “in the United States lawns cover more surface area — 50,000 square miles — than any single farm crop”, and “in the western United States about 60 per cent of all the water that comes out of taps for all purposes is sprinkled on lawns”. So if you are relatively wealthy you employ someone to look after your lawn. In effect, a gardener.

Lawn boys, like most domestic servants in the US, were and are usually (or at least often) either black or hispanic. Normally they are not literally “boys”, minors, but grown men. The word “boy” is a word, now regarded as demeaning and offensive, for a male servant:

“boy
6. Offensive A male servant, such as a valet.”
American Heritage Dictionary
http://education.yahoo.com/reference/dictionary/entry/boy

Moreover, in the southern US, the term “boy” is “an extremely offensive word used for talking to a black man, especially in the past”.
http://www.collinsdictionary.com/dictionary/english/boy

This demeaning use of “boy” for an adult black male servant is not confined to the US; it was also current British usage in colonial Africa, for example:

“(usually derogatory) (esp in former colonial territories) a Black person or native male servant of any age”
http://www.collinsdictionary.com/dictionary/english/boy

So if this writer had just put “lawn boys”, without “grown”, American readers at least would have understood that it meant adult male servants, not young boys. So why add “grown”? To understand this, I think we need to know that this passage comes from a novel by the African-American writer Toni Morrison, set in the American South in the 1950s. She is evoking the black “lawn boys” who tended rich white people’s lawns riding in the still segregated buses with the black maids who cleaned and cooked for the white folks and raised their children. Putting “grown” is a way of marking the fact that the so-called “lawn boys” are not boys, but men; it can be read as an implicit protest at the demeaning use of “boy” to refer to a black man: a way of distancing herself and us from the term, making us realise that in using it routinely we are subscribing to and perpetuating the racist attitude that lies behind it. It is as if she were saying “grown men subjected to the indignity of being called ‘boys’”.

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Note added at 4 hrs (2012-07-16 10:08:45 GMT)
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The archetype of the black "lawn boy" has not entirely disappeared. There is a 2008 film entitled Lawn Boy:

"A heart-warming, delightfully funny romantic comedy about a lawn boy who catches the eye of an upscale career woman. Together they must endure the pressures from their friends and family; which include his home-boys, his “Cheaters” watching sister, her snooty girlfriends, and her cocktail drinking, snobbish mother an father. Will their love carry them through it all?"
http://coolmoviezone.com/the-lawn-boy-2008/

This lawn boy is, of course, black. It's just a new version of an old cliché about the black "lawn boy" as an object of sexual desire, Lady Chatterley-style, for bored, neglected rich white suburban wives. It is a common scenario in pornography (as you can see by searching for "lawn boy" on the Internet). Similarly, black female servants (and formerly slaves) were traditional objects of sexual abuse from their white male employers (or owners).

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Note added at 4 hrs (2012-07-16 10:14:08 GMT)
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I hasten to add that the connotations just mentioned are not relevant to the passage we are considering.