All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
It is permissible for a Muslim to invoke Allah and seek refuge with Him by any of his Beautiful Names. Allah Says (what means): {Say, "Call upon Allah or call upon the Most Merciful. Whichever [name] you call - to Him belong the best names."} [Quran 17:110]
The Standing Committee for Scientific Research and Issuing Fataawa in the Kingdom of Saudi Arabia was asked: “Is it permissible for a man when he is seeking the help of Allah to say for example, ‘O Helper’, o Lord’; or when he is asking for the facilitation of his affairs: “O facilitator (of affairs); ‘O Lord? What is the condition in doing so? What is the ruling on those who say that, forgetfully, ignorantly, or deliberately?
The Standing Committee answered: “It is permissible for you to say what you mentioned, because what is intended by ‘O Helper’, ‘o Facilitator’, in your supplication, is Allah; as you explicitly stated in your statement: “O Lord, …..till your last supplication, whether you said that forgetfully, or ignorantly, or deliberately. May Allah give us success….” [End of quote]
However, it is not permissible to assign a specific mention or supplication that is not mentioned in the Sharee'ah for a specific need. Believing that it has any specification, such as repeating the verse that you mentioned in the question and believing that it has a specification in solving problems, or in helping in studies and so on, and your statement ya badee’ul ‘ajaa-ib bil khair ya badee’ in order to solve problems, then such a specification needs evidence from the Sharee’ah, as the Prophet stated some special mention of Allah and some special supplications at certain occasions, such as the supplication of the Prophet : “Allahumma la sahla illa ma ja‘ltahu sahlan wa anta taj‘alul-hazna sahlan itha shi't. (Which means: O Allah! Nothing is easy except what You have made easy. If You will, You can make the difficult easy).” [Ibn Hibbaan and others]
Hence, it is more appropriate to supplicate with the above supplication for rendering difficult things easy.
Also, you may supplicate with the supplication for releasing distress. The Prophet said: “The supplication of Thun-Noon (Prophet Yunus [Jonah] ) when he supplicated while he was in the belly of the whale was, 'La ilaaha illa anta subhaanaka inni kuntu min Ath-Thaalimeen.' (Which means: There is none worthy of worship except You, Glory to You, Indeed, I have been of the transgressors.) Indeed, there is absolutely no Muslim man who supplicates with it for anything, except that Allah responds to him.” [Ahmad and At-Tirmithi]
Many innovations first started with such specification that is not mentioned in the Sharee’ah, and it may be that the doer did not intend it to be a Sunnah, but soon it became a followed way and the one who does not do it is blamed.
However, restricting oneself to the supplications contained in the Quran and the Sunnah is better and more appropriate than innovated supplications that may involve some falsehood, or ambiguity or rough style, such as your saying, “ya badee’ul ‘ajaa-ib”. So it is better to supplicate with the supplications mentioned in the Sunnah, which we have mentioned earlier, as well as supplicating with what was mentioned in the hadeeth narrated by Ahmad and Al-Bukhaari: “…ya badi'us-samawaati wal-ardh, ya dhal-jalali wal-ikram! Ya hayyu ya qayyum! (….O Creator of the heavens and earth, O Possessor of majesty and honor, O Ever-living, O-Eternal, [I ask of You]…..)”
It should also be mentioned that it is not permissible to repeat the singular word, like when a supplicator or one who seeks help says: “Allah Allah Allah”
Ibn Taymiyyah said:
“As regards the "singular noun" like "Allah" " Allah", Or with the pronoun "He" "He" [which refers to Allah], then this is not permissible in the book of Allah or the Sunnah. It was also not said by any of the scholars of the Salaf nor said by individuals of this Ummah (nation), who are followed; rather, it is the misguided late comers who adopted it.
They might have followed the example of a misguided Shaykh, as it was reported that ash-Shibli used to say: " Allah Allah ". It was said to him: Why do you not say ‘laa ilaaha illa Allah’ (None has the right to be worshipped but Allah?) He replied: I am afraid to die between the negation and affirmation [i.e. between negating the existence of any other deity than Allah –which is the first part (laa ilaaha); and the affirmation, which is affirming the existence of Allah Alone]. This is one of the mistakes of ash-Shibli that will be forgiven for him, due to his true faith, and his strong rationality, and being overcome by the circumstances, it may be that he was possessed by Jinn and taken to a psychiatric hospital and shaves his beard.
He has many similar matters in which it is not permissible to imitate him in it, even if he is excused or rewarded, because if a person wants to say ‘laa ilaaha illa Allah’ and died before he completes it, then this does not harm him at all. Because actions are judged by the intentions; and what he had intended will be written for him.
Perhaps some people exaggerated about this to the point that he considers the mention of the singular name for the specific people [due to their high status], and the mention of the full expression for the public.
Perhaps some of them said: “laa ilaaha illa Allah’ (There is no God but Allah) for the believers and "Allah" for knowledgeable people, and "He" for the reviewers among the scholars. Perhaps one of them would limit himself when he is in seclusion to "Allah Allah’ " or "He" or "O He" or “There is no He except He" . Perhaps some Sufi authors mentioned this, that this is glorified. Some of them reported that the Prophet taught Ali ibn Abu Taalib to say: “Allah Allah Allah’’ and the Prophet said it three times. This is a fabricated Hadeeth according to the agreement of the scholars of Hadeeth.” [End of quote]
For more benefit, please refer to Fataawa 343738 and 343300.
Allah knows best.