Summary of answer:
Praise be to Allah
Muslim narrated in his Saheeh (399) that ‘Umar ibn al-Khattab used to recite these words out loud, saying: “Subhaanaka Allahumma wa bi hamdika, tabaaraka asmuka wa ta‘aala jadduka wa laa ilaaha ghayruka (Glory and praise be to you O Allah, blessed be Your name and exalted be Your Majesty; there is no god but You).”
This has been narrated in marfoo‘ reports (with isnaads going back to the Prophet (blessings and peace of Allah be upon him)) and in mawqoof reports in which the isnaad ends with ‘Umar or other Companions of the Prophet (blessings and peace of Allah be upon him).
See: as-Sunan by ad-Daaraqutni (2/58); Silsilat al-Ahaadeeth as-Saheehah (2996); Sifat Salaat an-Nabi (sall-Allahu ‘alayhi wa sallam), by al-Albaani (may Allah have mercy on him), p. 93
The scholars of hadith, fiqh, Arabic language and others have stated that the vowelling on this word is a fat-hah, so it is read as al-jadd; they did not mention anything else.
Al-jadd means majesty or greatness.
So what the hadith means is: Exalted be Your greatness.
An-Nawawi (may Allah have mercy on him) said in Tahdheeb al-Asmaa’ wa’l-Lughaat (3/331):
The phrase in du‘aa’ al-istiftaah, “Wa ta‘aalaa jadduka”, with a fat-hah on the jeem, mean: Exalted be Your greatness.
And it was said that what is meant by al-jadd is independence of means. Both meanings are good, although al-Khattaabi did not mention anything but greatness. Another example is the verse in which Allah, may He be exalted, says, speaking of the jinn (interpretation of the meaning):
“And [it teaches] that exalted is the nobleness [jadd] of our Lord”
Al-Ba‘li said in al-Matla‘ ‘ala Abwaab al-Maqna‘ (p. 46):
[It is to be pronounced as] “Wa ta‘aalaa jadduka”, with a fat-hah on the jeem. End quote.
Al-Bahooti (may Allah have mercy on him) said in Kashshaaf al-Qinaa‘ (2/478):
[It is to be pronounced as] “Wa ta‘aalaa jadduka”, with a fat-hah on the jeem; What it means is: exalted be Your greatness. End quote.
In Ibn Qaasim’s commentary on ar-Rawd al-Murbi‘ (2/22) it says:
The word al-jadd, with a fat-hah on the jeem, means greatness, fortune, happiness, and independence of means. The word ta‘aala (exalted be…) means: Great is His… This refers to His being emphatically great. It is indicative of the utmost and ultimate highness: Exalted be Your Majesty, exalted be Your greatness, which surpasses all other greatness; Your might supersedes all other might; Your dominion overrides all other dominion. End quote.
In Tawdeeh al-Ahkaam Sharh Buloogh al-Maraam by al-Bassaam (2/169), it says:
Jadduka, with a fat-hah on the jeem and a shaddah on the daal, means: Your greatness, Your might, and Your dominion. End quote.
This word – jadd – appears several times with a fat-hah on the jeem in the Holy Qur’an and the Prophetic Sunnah.
Allah, may He be exalted, tells us that the jinn said:
“And [it teaches] that exalted is the nobleness [jadd] of our Lord; He has not taken a wife or a son”
As-Sa‘di (may Allah have mercy on him) said (p. 890): That is, exalted be His greatness and sanctified be His names. End quote.
Al-Qurtubi (may Allah have mercy on him) said (19/8): “exalted is the nobleness [jadd] of our Lord” means: His greatness and His majesty. End quote.
The Prophet (blessings and peace of Allah be upon him) said: “O Allah, none can withhold what You give, and none can give what You withhold, and the good fortune [jadd] of any fortunate person is to no avail against You.” Narrated by al-Bukhaari (844) and Muslim (593).
The scholars said that what this means is: the good fortune, independence of means and status of the fortunate, independent and high-ranking person are all to no avail against You.
Ibn Rajab said concerning the words of the Prophet (blessings and peace of Allah be upon him), “and the good fortune [jadd] of any fortunate person is to no avail against You”:
What is meant by the word jadd, with a fat-hah on the jeem, in this hadith is: All that the person who is independent of means possesses of resources will not avail him against You.
End quote from Fat-h al-Baari by Ibn Rajab (7/417).
An-Nawawi said in Sharh Saheeh Muslim:
With regard to the phrase dhaa’l-jadd (translated here as fortunate person), the most well-known view is that there is a fat-hah on the jeem. This is the way it was vowelled by the earlier and later scholars.
Ibn ‘Abd al-Barr said: Some of them narrated it with a kasrah (jidd). Abu Ja‘far Muhammad ibn Jareer at-Tabari said: It is with a fat-hah. Ash-Shaybaani pronounced it with a kasrah, but this is contrary to what is known to those who narrated the reports. It is not known that anyone else said that.
At-Tabari and those who came after him regarded the reports which spell this word with a kasrah as da‘eef (weak). They said: Regardless of the weakness of the report, what this word (i.e., jidd) means is striving hard. In other words, the phrase would mean that the effort of one who strives hard will not avail him against You; all that could benefit him and save him is Your mercy.
And it was said that what is meant is the one who strives his utmost for worldly gain.
It was also said that what it means is hastening to flee; in other words, if a person hastens to flee, his haste in fleeing will not avail him, for He is within Your grasp and subject to Your control and dominion.
The correct and well-known view is that al-jadd, with a fat-hah, refers to good fortune, independence of means, greatness and dominion. What is meant is: if a person enjoys good fortune in this world, his wealth, sons, greatness and power will not avail him against You and his good fortune will not save him from You; rather what will benefit him and save him is righteous deeds, as Allah, may He be exalted, says (interpretation of the meaning):
“Wealth and children are [but] adornment of the worldly life. But the enduring good deeds are better to your Lord”
Thus it will become clear that the correct vowelling of the phrase “wa ta‘aala jadduka” is with a fat-hah on the jeem, according to scholarly consensus, and what is referred to is greatness.
It is not correct to pronounce it with a kasrah (jidduka); this is distorting the words of the Messenger (blessings and peace of Allah be upon him), because he (blessings and peace of Allah be upon him) pronounced it with a fat-hah and the scholars did not refer to any alternative pronunciation.
Moreover, pronouncing it with a kasrah changes the meaning of the hadith, because al-jidd (with a kasrah) means striving hard or seriousness, which is the opposite of frivolity. This is not what is meant in this hadith.
What anyone who wants to alert people to the mistakes they make must do is make sure of what he is saying before spreading it, especially if it has to do with Islamic rulings and prophetic hadith, so that he will not be telling lies inadvertently against the Messenger (blessings and peace of Allah be upon him), and so that he will not be telling them not to do what is right and will be telling them to do it the wrong way, in which case he will become one of “those whose effort is lost in worldly life, while they think that they are doing well in work” [al-Kahf 18:104].
And Allah knows best.