He is asking about the hadeeth, “The most beloved names to Allaah…”

The hadeeth referred to in the question is saheeh. It was
narrated by Muslim in his Saheeh (2132) from the hadeeth of Ibn
‘Umar (may Allaah be pleased with him) who said: The Messenger of Allaah
(peace and blessings of Allaah be upon him) said: “The most beloved
of your names to Allaah are ‘Abd-Allaah and ‘Abd al-Rahmaan.” 

Abu Dawood (4950) narrated that Abu Wahb al-Jushami, who was
a companion of the Prophet (peace and blessings of Allaah be upon him),
said: The Messenger of Allaah (peace and blessings of Allaah be upon
him) said: “Give your children the names of Prophets, and the most beloved
of names to Allaah are ‘Abd-Allaah and ‘Abd al-Rahmaan, and the most
truthful are Haarith and Hammaam [these names both refer to one who is
always doing something, whether good or bad], and the worst are Harb and
Murrah [meaning ‘war’ and ‘bitterness’ respectively].” Classed as saheeh by
al-Albaani in Saheeh Abi Dawood. 

The names which reflect
servitude to Allaah (those that start with “ ‘Abd” meaning “slave of”) are
more beloved to Allaah than others, because they affirm the attributes that
befit Him and do not befit anyone else. No one among mankind has any right
to them or any share in them. This has to do with His divine Attributes
(sifaat al-uloohiyyah). They also affirm the status that befits the person,
which never alters for an instant throughout his lifetime, namely the status
of servitude, of being the slave of his Lord. So the name reflects well on
the one who carries it; it honours the person who is given this name by
describing him as a slave of his Lord. Hence these names carry great
virtue. 

The connection between the slave and Allaah is one of pure
servitude (‘uboodiyyah), and the connection between Allaah and the slave is
one of complete mercy. It is by His mercy that he exists in the true sense
of the word, and the purpose for which he was created is to worship Allaah
alone, with love, fear, hope, worship and veneration, so he is a slave to
Allaah. 

This is what is meant by the name: that the owner of this
name is connected to one of the attributes of Allaah which cannot be
attributed to anyone else. 

Because His mercy prevails over His wrath, mercy is more
beloved to Him than anger, so the name ‘Abd al-Rahmaan (“slave of the Most
Merciful”) is dearer to him than the name ‘Abd al-Qaahir (“slave of the
Subduer”) and similar names. 

So these names refer to these two meanings, that of divinity
and that of servitude, and that means that the name is a constant reminder
of the slave’s humble position before his Lord, and a constant prayer for
the Lord’s mercy towards His poor, humble servant. 

Some scholars have stated that these two names have their own
special features, which is that in the Qur’aan, Allaah does not mention the
word ‘abd (or its plural ‘ibaad) in conjunction with any of the names of
Allaah except these two. Allaah says (interpretation of the meaning): 

“And when the slave of Allaah [‘abd Allaah] (Muhammad)
stood up invoking Him”

[al-Jinn 72:19] 

“And the (faithful) slaves of the Most Gracious [‘ibaad
al-Rahmaan] (Allaah)”

[al-Furqaan 25:63] 

This is supported by the verse in which Allaah says
(interpretation of the meaning): 

“Say (O Muhammad): Invoke Allaah or invoke the Most
Gracious [al-Rahmaan] (Allaah)”

[al-Isra’ 17:110] 

If the first thing of which a child becomes aware and
understands and hears is this name, the idea will take root in his heart
that he is a slave of Allaah and that Allaah is his Lord, and that his Lord
is the Most Gracious, Most Merciful, and it will be easy to raise him
accordingly. 

For more information please see Tuhfat al-Mawdood bi
Ahkaam al-Mawlood, by Ibn al-Qayyim; Fath al-Baari by Ibn Hajar,
in his commentary on the chapter Ahabb al-Asma’ ila Allaah (the most beloved
names to Allaah); and al-Sindi’s commentary on Ibn Maajah (Haashiyat
al-Sindi ‘ala Ibn Maajah). 

May Allaah help us and
you to be sincere slaves to Him, and admit us to His Mercy by His grace and
bounty, for He is the most merciful of those who show mercy.