There is nothing wrong with holding this kind of competition, because it is spreading goodness, benefiting all concerned, and increasing knowledge, especially if the participant writes the meaning of the name that he mentions. But it is important to use sound books for this information, and to beware of quoting from people of misguidance and deviation.
The one who wants to adhere to the names of Allaah that are proven in the Qur'aan and Sunnah is advised to refer to al-Qawaa’id al-Muthla by Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) and Sifaat Allaah ‘azza wa jall al-Waaridah fi’l-Kitaab wa’l-Sunnah by Shaykh ‘Alawi al-Saqqaaf (may Allaah preserve him).
The members should be encouraged to memorize these names and pay attention to them, because of the report narrated by al-Bukhaari (2736) and Muslim (2677) from Abu Hurayrah (may Allaah be pleased with him), according to which the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Allaah has ninety-nine names, one hundred less one. Whoever memorizes them will enter Paradise.”
The memorizing mentioned in the hadeeth includes the following:
1. Learning them by heart.
2. Learning their meanings.
3. Acting upon their meanings. So if a person learns that He is al-Ahad (the One), he should not associate anyone else with Him. If he learns that He is al-Razzaaq (the Provider), he should not seek provision from anyone else. If he learns that He is al-Raheem (the Most Merciful), he should not despair of His mercy, and so on.
4. Calling upon Him by them, as He says (interpretation of the meaning): “And (all) the Most Beautiful Names belong to Allaah, so call on Him by them” [al-A’raaf 7:180]. That is by saying for example: Ya Rahmaan irhamni (O Most Merciful, have mercy on me), Ya Ghafoor ighfir li (O Oft Forgiving, forgive me), Ya Tawwaab tubb ‘alayya (O Accepter of repentance, accept my repentance) and so on.
This competition does not come under the heading of communal dhikr which is an innovation, because that is recited in unison, which is not the case here.
For more information please see the answer to question no. 111562.
And Allaah knows best.