It is possible to use the stream of thoughts itself as the concentration object, but I've found that geeks and nerds tend to attach (and value) too much to the thoughts themselves. Call this, "getting lost in stories."

I say this because there is a surrender when you are being mindful. You are surrendering the thought -- rather than surrendering to the thought -- yet somehow, that which witnesses -- experiences -- the thought remains without effort. I used to think, being mindful requires a sort of force, a way of isolating what you are doing here and now. That does not work, a good way to make your mind rigid enough to shatter. And thankfully, not something most people can sustain for long periods of time without becoming exhausted.

When you start meditating, a stream of thought shows up like a train wreck and you get carried along with it. You get lost in stories. Stories and narratives about this and that, why you should do this or that, why this thing happened or that person said this. Making up interpretations of things. Correcting and analyzing your practice as you are practicing. Very comfortable for nerds -- very comfortable for most people -- but still lost in stories. That's mindlessness.

In the deeper states, the "stories" you get lost in start taking on qualities of lucid dreams. That's why there are Zen koans like:

A student was meditating and saw visions
of horrors and hell. Terrified, he ran to
his teacher. The teacher listened and told
him, "Let it go."

A few days later, the student was meditating
and saw visions of heaven. He was blissing out,
totally high as waves of pleasure ran through
him. Suddenly, he felt his teacher hit him on
the head. "Let it go."

This "train wreck" of a thought stream, once going on, is very difficult to suppress. By that point, you don't want to suppress it. It's more of riding it out because in time, that too shall pass.

That's also why you have quotes from say, the Tao Te Ching, "A journey of ten thousand miles begin with the single step." Americans and the Chinese love this quote, often taking it to mean that to achieve great things, you must start with that first step. It's just that, at a different level, it is also teaching you how to not get lost in stories. The verse runs like this:

It's easy to keep hold of what hasn't stirred,
easy to plan what hasn't occurred.
It's easy to shatter delicate things,
easy to scatter little things.
Do things before they happen.
Get them straight before they get mixed up.

The tree you can't reach your arms around
grew from a tiny seedling.
The nine-story tower rises
from a heap of clay.
The ten-thousand-mile journey
begins beneath your foot.

(Verse 64, trans. by Ursula K Le Guin)

It is very easy to take that ten-thousand-mile journey and get lost in your stories. Stuff of flame wars and new operating systems :-)

It is also noting that, as your powers of observation and concentration grows, you will be able to observe these thoughts, emotions, and sensations closer to when they are "born". And releasing them before they become something strong enough that you can only ride out its fury.

There's another side to this. Part of mindfulness, that is, of insight practice (rather than concentration practice), is to probe into the artifacts of your mind. For a stream of thought, you can observe what constitute "thought." You probe into the underlying emotions and the symphony of sensations. If you pay attention, you'll find that emotions and physical sensations precede the "thought", and that the "stream of thoughts" is not as real or tangible as you normally experience it.

See through that bullshit long enough, you start noticing a lot of things in your day to day life are simply BS. Or how you, and no one else, are the one who made yourself unhappy, or fearful, or angry, or sad. Easy to observe when you're sitting still, much harder when you are talking to flesh-and-blood people. Actually applying what you learned on the cushion into day to day life will take lifetimes, more difficult than anything else you will ever learn, and more fun and adventurous.