As far as I know, the Google Suggestions are based on commonly made query revisions. In other words, they come "from the universe of all searchers" and there's little - maybe nothing - we can do about it.

I've got one such situation and what I'm trying to do for the client is rank well on the suggested query - at least getting the client's point of view and messaging seen. I've got three results on the bottom of page 1 now, and of course I'm working to improve on that.

I had a week of hope in early October when the identical "related searches" entry vanished from the bottom of the SERP - but it came back, and the Google Suggestion never went away.

I experienced this recently while searching for a relatively obscure silversmith by name. I spelled the surname incorrectly on a number of searches...then I called someone to ask how to spell the guys name...I was one letter out...but Google’s suggestion kept popping the incorrect name up as a suggestion. So that certainly got me thinking how easy it might be for a blackhat to repeatedly use a term to discredit a competitor, eg [business name] complaints, [business name] non delivery, [business name] theft, [business name] plagiarism etc. I cleared my cache a number of times, but a week later google was still "suggesting" the silversmith's name the way I had incorrectly spelt it.

It got me worried...

So I guess the whitehat way is to push it down, and the blackhat way would be to artificially inflate the number of searches for other terms? The company name doesn't get a lot of searches, so I'm guessing beating out "[business name] complaints" in # of searches shouldn't be an insurmountable task with some clever tricks. By comparison trying to take over the first page for "[business name] complaints" would be a nightmare...not to mention there's also "[business name] scam" to deal with :P

Luckily so far the related searches at the bottom are all neutral.