Praise be to Allah

Firstly:

It was narrated from ‘Aa’ishah (may Allah be pleased with her) that when the Messenger of Allah (blessings and peace of Allah be upon him) saw rain, he would say: “Allahumma sayyiban naafi‘an {O Allah, (make it) a beneficial downpour).” Narrated by al-Bukhaari (1032)

According to a version narrated by Abu Dawood (5099), he used to say: “Allahumma sayyiban hanee’an (O Allah, (make it) a wholesome downpour).” Classed as saheeh by al-Albaani.

The word sayyib (translated here as downpour) refers to rain that flows. Allah, may He be exalted, says (interpretation of the meaning): “Or like a heavy downpour [sayyib] from the sky” [al-Baqarah 2:19].

See: Ma‘aalim as-Sunan by al-Khattaabi (4/146)

It is mustahabb to expose oneself to the rain and let some of it flow over one’s body, because of the proven report from Anas (may Allah be pleased with him) who said: When we were with the Messenger of Allah (blessings and peace of Allah be upon him) it rained. The Messenger of Allah (blessings and peace of Allah be upon him) lifted part of his garment so that the rain could fall on him. We said: O Messenger of Allah, why did you do that? He said: “Because it has just come from its Lord, may He be glorified and exalted.” Narrated by Muslim (898).

When the rain grew severe, the Prophet (blessings and peace of Allah be upon him) said: “Allahumma hawaalayna wa la ‘alayna; Allahumma ‘ala al-aakaami wa’z-ziraabi, wa butoon al-awdiyah wa manaabit ash-shajar (O Allah, (let the rain fall) around us and not upon us, O Allah, (let it fall) the small mountains and hillocks, the valley bottoms and places where trees grow).” Narrated by al-Bukhaari (1014).

With regard to du‘aa’ when hearing thunder, it is proven from ‘Abdullah ibn az-Zubayr (may Allah be pleased with him) that when he heard thunder he would stop speaking and would say: Subhaan alladhi yusabbih ar-ra‘du bi hamdihi wa’l-malaa’ikatu min kheefatihi (Glory be to the One whom the thunder glorifies and praises, and so do the angels because of His Awe) [cf. ar-Ra‘d 13:13]. Then he would say: This is a stern warning to the people of earth. Narrated by al-Bukhaari in al-Adab al-Mufrad (723); Maalik in al-Muwatta’ (3641). Its isnaad was classed as saheeh by an-Nawawi in al-Adhkaar (235) and by al-Albaani in Saheeh al-Adab al-Mufrad (556).

We do not know of any marfoo‘ report from the Prophet (blessings and peace of Allah be upon him) concerning that.

Similarly, there is no proven dhikr or du‘aa’ from the Prophet (blessings and peace of Allah be upon him) for seeing lightning, as far as we know. And Allah knows best.

Secondly:

The time when rain comes down is a time of divine bounty and mercy from Allah to His slaves, when the means of goodness are abundant, and it is a time when it is thought that du‘aa’s will be answered.

It says in the marfoo‘ hadith of Sahl ibn Sa‘d that the Prophet (blessings and peace of Allah be upon him) said: “Two (du‘aa’s) are not rejected: du‘aa’ at the time of the call to prayer and du‘aa’ at the time of rain.”

Narrated by al-Haakim in al-Mustadrak (2534); at-Tabaraani in al-Mu‘jam al-Kabeer (5756); classed as saheeh by al-Albaani in Saheeh al-Jaami‘ (3078).

And Allah knows best.