Is it mustahabb to recite a specific du‘aa’ when kissing the Black Stone?

The Sunnah when touching or kissing the Black Stone when
starting tawaaf is to say: 

“Bismillah, wa Allahu akbar (in the name of Allah, and
Allah is most great).” If one adds “Allahumma
eemaanan bika wa tasdeeqan bi kitaabika wa wafaa’an bi ‘ahdika wa ittibaa‘an
li Sunnati Nabiyyika Muhammad sall-Allahu ‘alayhi wa sallam
(O Allah, out of faith in You, and in belief in Your Book, and in fulfilment
of Your covenant, and following the Sunnah of Your Prophet Muhammad
blessings and peace of Allah be upon him)”,
there is nothing wrong with that. 

The evidence for that is the report narrated from Ibn ‘Abbaas
(may Allah be pleased with him), according to which the Messenger of Allah
(blessings and peace of Allah be upon him) circumambulated the Ka‘bah on a
camel, and every time he came to the corner [where the Black Stone is], he
pointed to it with something in his hand and said takbeer.

Narrated by al-Bukhaari (1632). 

It was narrated by ‘Abd ar-Razzaaq in al-Musannaf
(5/33) from Ibn ‘Umar (may Allah be pleased with him) that when he touched
the corner [where the Black Stone is] he said: Bismillah, wa Allahu akbar
(in the name of Allah, and Allah is most great).

Classed as saheeh by al-Haafiz Ibn Hajar in at-Talkhees
al-Habeer (2/247). 

It was narrated from Ibn ‘Abbaas (may Allah be pleased with
him) that when he touched [the Black Stone], he said: “Allahumma
eemaanan bika wa tasdeeqan bi kitaabika wa Sunnati Nabiyyika Muhammad
sall-Allahu ‘alayhi wa sallam
(O Allah, out of faith in You, and in belief in Your Book and the Sunnah of
Your Prophet Muhammad blessings and peace of Allah be upon him)”. 

A similar report was narrated by al-Bayhaqi in as-Sunan
al-Kubra (5/79) from ‘Ali ibn Abi Taalib (may Allah be pleased with him).
These reports were also narrated by at-Tabaraani. 

Many of the fuqaha’ and scholars stated that this dhikr is
mustahabb. 

See: al-Umm by ash-Shaafa‘i (8/163); al-Majmoo‘
by an-Nawawi (8/49); al-Mughni by Ibn Qudaamah (3/183); Fath al-Qadeer
by Ibn Humaam (2/448) 

Shaykh Ibn Baaz (may Allah have mercy on him) said: 

It is prescribed to say Takbeer when touching or kissing the
Black Stone, and when pointing to it if it is not possible to touch it.

End quote from Majmoo‘ Fataawa Ibn Baaz (17/223). 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: 

What should be said when touching the Black Stone? 

When touching it for the first time, one should say:
‘Bismillah
wa Allahu akbar. Allahumma eemaanan bika wa tasdeeqan bi kitaabika wa
wafaa’an bi ‘ahdika wa ittibaa‘an li Sunnati Nabiyyika Muhammad sall-Allahu
‘alayhi wa sallam (In the name of Allah, and Allah is most great. O
Allah, out of faith in You, and in belief in Your Book, and in fulfilment of
Your covenant, and following the Sunnah of Your Prophet Muhammad blessings
and peace of Allah be upon him)..
Then when one has gone around and come back in line with the Black Stone,
one should say “Allahu akbar” only. 

End quote from al-Liqa’ ash-Shahri (no. 10). 

And Allah knows best.