How to improve mindfulness of thinking?

THE 18 INVESTIGATIONS. 'Bhikshu, this person is made up of the eighteen mental investigations';
so it is said. And in what connection is this said?
i. On seeing a form with the eye,
   one investigates the form that is the basis for mental joy,
   one investigates the form that is the basis of mental pain,
   one investigates the form that is the basis of equanimity.
ii. On hearing a sound with the ear,
   one investigates the sound that is the basis for mental joy,
   one investigates the sound that is the basis of mental pain,
   one investigates the sound that the basis of equanimity.
iii. On smelling a smell with the nose,
   one investigates the smell that is the basis for mental joy,
   one investigates the smell that is the basis of mental pain,
   one investigates the smell that is the basis of equanimity.
iv. On tasting a taste with the tongue,
   one investigates the taste that is the basis for mental joy,
   one investigates the taste that is the basis of mental pain,
   one investigates the taste that is the basis of equanimity.
v. On feeling a touch with the body,
   one investigates the touch that is the basis for mental joy,
   one investigates the touch that is the basis of mental pain,
   one investigates the touch that is the basis of equanimity.
vi. On cognizing a mind-object with the mind,
   one investigates the mind-object that the basis of mental joy,
   one investigates the mind-object that is the basis of mental pain,
   one investigates the mind-object that is the basis of equanimity.
'Bhikshu, this person is made up of the eighteen mental investigations': so it is said in this connection.

Dhātu Vibhaṅga Sutta similarly in Titth’ayatana Sutta

Essentially there are 3 sensation which arise due to metal activity: pleasant, unpleasant and neutral which gives arise to the unwholesome roots. It is by looking at these sensations with equanimity and rightly seeing their impermanence you can prevent them from arising. [Pahāna Sutta, Avijja Pahana Sutta 2]

Being mindful when thinking you should not have the unwholesome roots to arise. If it does this leads to unwholesome action, hence unpleasant results in the future. [Mula Sutta] If your thinking is without the unwholesome roots then there is no future unpleasant results. This is what you should aim for in being mindful while you are thinking, i.e., do not let the unwholesome roots arise, do not let the hindrances arise, do not let the Vipallasa arise. This can be achieved by being equanimous and watching the arising and passing of phenomena. Having Wise Attention all 8 Path Factors develop automatically. [Yoniso Manasikāra Sampadā Sutta]

I'd honestly say to focus on your breath, then just allow thought to come and go naturally.

If you have trouble "noting" your thoughts or thought process--try instead to focus on the feeling that the thought produces.

Remember though, the idea of noting thoughts is one of a very soft touch, never forcing yourself to think about your thinking. Quite simply acknowledging that a thought exists or has passed is more than enough.

Everything else depends on time and practice. And try not to try so hard. :)

I don't think this is a sign of weak mindfulness. Actually, the opposite might be the case: When your mindfulness improves you'll be able to see more clearly. But seeing more cleary often means seeing the mess and chaos in mind and body that is going on all the time whether we are aware of it or not.

Moreover, you're describing the 3 characteristics:

• Thoughts are impermanent and change quickly
• You cannot control when a thought comes up, nor can you really control your awareness of thoughts
• Thoughts are not satisfying. You say not being aware of thinking is a problem for you. Who was first, the thought or you not being a aware? (Not) wanting and therefore suffering can arise from thoughts.

I wouldn't worry about it too much. Wherever you catch thinking that's where you should note it. If you realize you were not mindful you can note it with 'knowing'. Actually when you realize you were not mindful before, that's the only way of becoming mindful again, by becoming mindful of not being mindful.

But having a stable eye is really a wrong perception of what meditation is about. It's supposed to be unstable.

My understanding is the 3rd frame of reference (cittanupassana) does not involve observing thoughts (vitakka). It involves knowing whether the mind-heart (citta) is defiled or not. Defilements (kilesa) are not thoughts. Although there can be 'defiled thoughts', 'defilements' (kilesa) are the energy/drive/mood of greed, hatred & delusion, which are not thoughts.

For example, when there is sexual lust, anger or fear, the energy or vibrations that runs through & simulates the physical body with lust, anger or fear, giving rise to all kinds of physical reactions & tensions, is not 'thought'.

The teaching about the 3rd frame of reference does not mention thoughts. It states:

And how does a monk remain focused on the mind in & of itself? There is the case where a monk, when the mind has lust, discerns that the mind has lust. When the mind is without lust, he discerns that the mind is without lust. When the mind has hatred, he discerns that the mind has hatred. When the mind is without hatred, he discerns that the mind is without hatred. When the mind has delusion, he discerns that the mind has delusion. When the mind is without delusion, he discerns that the mind is without delusion.

When the mind is constricted, he discerns that the mind is constricted. When the mind is scattered, he discerns that the mind is scattered. When the mind is enlarged, he discerns that the mind is enlarged. When the mind is not enlarged, he discerns that the mind is not enlarged. When the mind is surpassed, he discerns that the mind is surpassed. When the mind is unsurpassed, he discerns that the mind is unsurpassed. When the mind is concentrated, he discerns that the mind is concentrated. When the mind is not concentrated, he discerns that the mind is not concentrated. When the mind is released, he discerns that the mind is released. When the mind is not released, he discerns that the mind is not released.

Satipatthana Sutta

To be able to observe clearly or meticulously the arising, existing & passing of thoughts requires a high degree of concentration (mental collectedness/clarity/stability). The teachings state:

And what is the (mental) development using concentration (samādhi) that, when developed & pursued, leads to mindfulness (sati) & clear-comprehension (sampajaññā)? There is the case where feelings (vedanā) are known to the monk as they arise, known as they persist, known as they subside. Perceptions (saññā) are known to him as they arise, known as they persist, known as they subside. Thoughts (vitakkā) are known to him as they arise, known as they persist, known as they subside. This is the (mental) development using concentration that, when developed & pursued, leads to mindfulness & clear-comprehension.

Samadhi Sutta

A tip I can provide is to develop concentration (samadhi), such as by using Mindfulness With Breathing (Anapanasati).

An analogy is as follows. If you wish to scientifically examine water, ideally, you take a drop of water & place it under a microscope. You do not walk into a raging surf (in the ocean) with a microscope. Similarly, to observe thoughts in a meticulous manner generally requires a large amount of concentration and a small amount of thoughts.