I often hear people say, "hm, I'll go meditate on that..." Or a guy I know who says he meditates on his ideas for his art. But actually, that is the complete opposite of what meditation is supposed to be. When properly meditating, the goal is to not concentrate or think about anything. I just point this out because the basics of what meditation actually is are widely misunderstood by most Westerners -- and even some…
Your (mis)understanding (that meditation is spacing out and "not thinking") seems to be the common western one. In my experience, this is usually due to misunderstood ideas about practices like dzogchen or philosophical concepts like dependent origination.
Concentration is actually at the _heart_ of meditation and is a key element of the Buddhist noble eight-fold path.
Mindfulness is merely well-concentrated investigation. Once someone has developed an ability to cultivate sustained, one-pointed attention - they can use this attention to clearly investigate internal and external phenomena.
Many Asian schools have the first four Samatha Jhanas (deeply-concentrated states) as strict prerequisites for beginning Vipassana (insight / mindfulness) practice. Others regard "access concentration" as the only prerequisite.
In any case, a well-concentrated mind is required to clearly observe oneself. Yes, other thoughts will drop away as the mind naturally gravitates toward the joy and rapture of mental seclusion (from disturbing, burdensome thoughts and mental states) - but this is a _byproduct_ of concentration, not a fruit of _avoiding_ concentration.
The concentration states (along with the Brahma Viharas) can also be welcome refuges for when reality starts to come apart as one progresses through the Vipassana Nanas.
These things are very clearly spelled out in the canonical Buddhist texts and especially the commentaries[1][2]. Westerners seeking to "change their brains" do so at their peril without proper guidance. Peeling away the layers of delusion can be a destabilizing process for some[3].
Bhante Henepola Gunaratana's book "Mindfulness in Plain English" is a very accessible (and worthwhile) introduction to the subject[4].
[1] - [pdf] http://www.accesstoinsight.org/lib/authors/nanamoli/PathofPu...
[2] - [pdf] http://urbandharma.org/pdf1/Path_of_Freedom_Vimuttimagga.pdf
[3] - http://www.theatlantic.com/health/archive/2014/06/the-dark-k...
[4] - http://www.amazon.com/Mindfulness-English-Bhante-Henepola-Gu...