I dont think that there are legitimate ways to influence suggest for popular keywords. I have noticed one thing though: Popularity of a particular term leads to its inclusion in the suggest list. Example: I ran an awareness/social campaign to save our historical monuments from vandalism by making a website where people could scribble whatever they want.
That campaign went off really well with retweets and shares among good influencial folks. It got shared on bookmarking websites as well. Suddenly, I started seeing a keyword "responsible travel" coming up on suggestions. But as the momentum died, we lost that preference. Maybe the QDF algorithm kicked that keyword out?