The phrase as used by most LDS folks comes from D&C 9:9. This section addresses Oliver Cowdery and his attempt to translate. Prior to this verse below he was admonished for "taking no thought save it was to ask". He took no thought, but his lips apparently asked to be able to translate. Then the admonition was to study it out in his mind. Then vs 9:
9 But if it be not right you shall have no such feelings, but you shall have a stupor of thought that shall cause you to forget the thing which is wrong; therefore, you cannot write that which is sacred save it be given you from me.
Stupor: a state of near-unconsciousness or insensibility
With that scripture, and the above definition from the dictionary D&C 9 takes on new meaning. Oliver was to recognize that which was not right by the dulled, suspended senses of his mind. The impaired consciousness.
Going to the Book of Mormon we get more information about this phrase. Here Moroni is rebuking Pahoran over what appears to be the government’s tragic neglect of Moroni's armies
Alma 60:6 And now behold, we desire to know the cause of this exceedingly great neglect; yea, we desire to know the cause of your thoughtless state.
7 Can you think to sit upon your thrones in a state of thoughtless stupor, while your enemies are spreading the work of death around you? Yea, while they are murdering thousands of your brethren—
The tone of this verse gives a very "negligent" feel to the stupor. This state of thoughtless stupor doesn't mean the people have no brain activity. If you hooked them up to an EEG machine you may possibly see what scientists would call "normal" brain activity. It seems to more refer to the quality or intelligence (or lack of) of the mind. Or that they are neglecting something of supreme importance and are darkened. Joseph Smith warned what "neglecting" individual duty would bring: A darkening of the mind.
“President Joseph Smith read the 14th chapter of Ezekiel – said the Lord had declared by the Prophet [Ezekiel], that the people should each stand for himself, and depend on no man or men in that state of corruption of the Jewish Church – that righteous persons could only deliver their own souls – applied it to the present state  of the Church of Jesus Christ of Latter-day Saints – said if the people departed from the Lord, they must fall – that they were depending on the Prophet, hence were darkened in their minds, in consequence of neglecting the duties devolving upon themselves...”
- The Teachings of the Prophet Joseph Smith, pp. 237-238
I've concluded a stupor of thought is the opposite of enlightenment. It's a state of mind which affects your soul. Not just temporary, but one which can last years. Related terms might be a "darkening" of the mind or "sleep" due to laboring under false ideas and false beliefs which cause a person to stagnate and carry on in a thoughtless neglectful stupor. Miserable and bored, yet not quite aware of it. It's brought on by neglect of God. In preference to idols, laziness, gluttony, and false and fallen ideas.
7 Behold, he changed their hearts; yea, he awakened them out of a deep sleep, and they awoke unto God. Behold, they were in the midst of darkness; nevertheless, their souls were illuminated by the light of the everlasting word; yea, they were encircled about by the bands of death, and the chains of hell, and an everlasting destruction did await them.
It's sometimes difficult to awaken someone out of such a stupor or slumber. Moroni's rebuke was calculated to accomplish this. Pahoran proved he was already awake. Sometimes a rebuke, or showing someone a blaring contradiction can work. Sometimes gently and persuasively going through a line of reasoning that creates a new picture can be effective. But sometimes this can go against their understanding and presents such a contradiction in the mind that ironically it gets labeled a "stupor" and is dismissed. As with physical sleep, often folks resent and do not want to be awakened from deep sleep.
This is the thoughtless stupor I think plagues us in 2016. We sometimes read about plagues in the last days and presume that exclusively refers to insects, or physical disease epidemics. However I think there are other very real plagues, which exist here and now, which simply go unnoticed.
I think Alma shows the way out. It involves a change of heart, and an awakening. In verse 7 quoted above Alma says "he" changed their hearts. Meaning God brought about this change. They were illuminated by the light of the everlasting word. The "everlasting word" of course is Christ. We must come to, as well as come to know Christ. The glory of God is intelligence, or in other words light and truth. D&C 93:36.
The word of God illuminates the soul. It doesn't just present new ideas, it illuminates the soul. It's one way to determine if a message comes from God. Those who pretend to speak for God, but which do not, will keep you in a thoughtless darkened stupor.
Consider the messages found at this Conference focused on the Doctrine of Christ. Test, and see if it does not awaken, encourage, and illimuniate. www.doctrineofchristconference.com