Praise be to Allah.
The hadith mentioned is saheeh; it was narrated by Muslim in his Saheeh (2698) from Mus‘ab ibn Sa‘d ibn Abi Waqqaas: My father told me: We were with the Messenger of Allah (blessings and peace of Allah be upon him) and he said: “Is it too difficult for any one of you to earn one thousand hasanahs (good deeds) every day?” One of those who were sitting with him asked him: How can one of us earn one thousand hasanahs? He said: “If he says one hundred tasbeehs, one thousand hasanahs will be recorded for him, or one thousand bad deeds will be erased for him.”
An-Nawawi said in al-Adhkaar (p. 53): Imam al-Haafiz Abu ‘Abdillah al-Humaydi said: This is how it appears in the book of Muslim in all reports: “or one thousand bad deeds will be erased for him.” Al-Barqaani said: It was narrated by Shu‘bah, Abu ‘Awaanah and Yahya al-Qattaan, from Moosaa, from whom Muslim narrated it. They said: “and one thousand bad deeds will be erased for him.” End quote.
‘Abdullah ibn Ahmad ibn Hanbal said: My father said: Ibn Numayr also said: “or one thousand bad deeds will be erased for him.” Ya‘laa also said: “or one thousand bad deeds will be erased for him.” See: Musnad Ahmad, ar-Risaalah edn. (3/133).
At-Tirmidhi narrated it as “and one thousand bad deeds will be erased for him” and said: [It is] hasan saheeh.
Al-Qaari said in Mirqaat al-Mafaateeh (4/1594): Because one good deed brings a tenfold reward, and this is the minimal multiplication promised in the Qur’an, in the verses (interpretation of the meaning):
“Whoever comes [on the Day of Judgement] with a good deed will have ten times the like thereof [to his credit]”
“Allah gives increase many times over to whom He pleases”
Good deeds done in the Haram [Makkah] bring a hundred thousandfold reward. “or one thousand bad deeds will be erased for him”, either minor or major; that is subject to the will of Allah, may He be exalted. End quote.
Based on that, whoever recites tasbeeh more than one hundred times will also be granted a multiple reward for whatever extra he does, because each good deed brings a tenfold reward. So whoever glorifies Allah one thousand times will attain ten thousand hasanahs, and so on. The bounty of Allah is immense.
Similar to this hadith is the report narrated by al-Bukhaari (3293) and Muslim (2691) from Abu Hurayrah (may Allah be pleased with him), according to which the Messenger of Allah (blessings and peace of Allah be upon him) said: ““Whoever says: ‘Laa ilaaha ill-Allaah wahdahu laa shareeka lah, lahu’l-mulk wa lahu’l-hamd wa huwa ‘ala kulli shay’in qadeer (There is no god but Allah alone with no partner or associate, His is the dominion, to Him be praise and He has power over all things)’ one hundred times in a day, it will be the equivalent of his freeing ten slaves, and one hundred good deeds will be recorded for him, and one hundred bad deeds will be erased for him, and it will be a protection for him against the shaytaan all that day until evening comes, and no one will do anything better than what he has done except one who does more than that.”
Muslim (2692) also narrated that Abu Hurayrah said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever says in the morning and in the evening: ‘Subhaan Allaah wa bi hamdihi (Glory and praise be to Allah)’ one hundred times, no one will come on the Day of Resurrection with anything better than what he has done, except one who said something like what he said, or more than that.”
Thus he stated that whoever does more than that has done better than the one who limited it to one hundred times, so he may recite that dhikr two hundred times a day, or three hundred times, or whatever Allah wills for him. Whoever does more, then that which is with Allah (of reward) is greater, and the bounty of Allah is immense.
For more information, please see the answer to question no. 220345.
And Allah knows best.