Anything without the name RAMA has no value
Name Rama is equivalent to the Omkar naadbrahma. When you recite your soul is ready to have mukti.
Ram nam Satya hai
Satya bolo mukti hai
There is a chance that the Atman(soul) of the particular Jeeva(person) which has just passed away will get free from the sansaar chakra(World), if recited by others who are performing last rituals. Hence this is like a giving best wishes to the atma(soul) of the dead person. The continuous chattering in a loop implies that the dead body is no longer a truth (has no breath) which is the name RAMA. Anything without the name RAMA has no value, whatsoever. In Ramayana, Hanumanji also broke mala given by sita because Rama's name was not written on it.
Glory of Rama name in Sages and scriptures
According to Gita 9th Adhyay If we speak name of Lord, we succeed in individual and universal blossoming.
Rama Nama gets exactly the same importance in Saguna Upasana (worshipping the Absolute Truth as a Being with a form) as does the Pranava ('Om') Mantra in Nirguna Upasana (worshipping the Absolute Truth as formless).
Rishi Manu shows that the Rama Nama has links to all the other mantras that ever exist.
In Syavana Smriti (Syavana is the name of a rishi), Syavana says that the Rama Nama is the essence of all the Vedas, Shastras, Itihasas and Puranas.
The chief among the Mantras for the Shaivites is the Panchakshari (5 lettered) mantra, 'Om Nama Shivaya'. Without the letter ‘ra’, ‘Narayana’ would read as ‘Nayana’ that means one without a way (‘gati’); without the letter ‘ma’, ‘Namasivaya’ would read as ‘Nasivaya’ that means inauspicious.
So the jiva akshara (life giving letters) 'Ra' and 'Ma' of these two Mantras are put together to form the Rama Nama!
Anyone chanting this Divine Name thus worships both Shiva and Vishnu at the same time.
Connection with Yoga
From source namadwaar:
In the philosophy of the Yoga, 'Ra' is considered the 'Mooladhara' (the point of origin of the power of Kundalini) and 'Ma' is the 'Sahasrara' (the destination). Hence, when the Rama Nama is chanted in the right tone and manner, the serpent power rises and hits the head or the 'kapala' Thus, by merely chanting the Rama Nama, one can become a Yogi.
So people speaks it for dead person's soul who is connected to the origin (Supreme soul) .
Another story related to Rama name,
Before writing Ramayana Rishi Valmiki was a thief and pirate named as " Valiya lutera". Sage Narada suggested him to chant Mara Mara whose opposite is Rama Rama. Valmiki started and he became sage out of the pirate and Lord Rama blessed him to write Ramayana.
It is truth that by chanting name of Rama everything can be accomplished. (This is my own experience that the things I deserve, if I do not get, by chanting the holy word Rama, I am always able to get.)
Mahatma Gandhi united the whole india by chanting of Rama only, and Lord Hanuman is worshiped in Hanuman Chalisa to have
Rama Rasayana means Rama names chemical that is generated from brain because of Rama name chanting and create power in the body.
Rama name itself is a truth, and when person dies he goes from Mriyu lok to Satya lok, so chanting of Rama's name gives peace to soul, even Ravana said Rama's name when he died!!!
Rama name is the home for people like me.
Rama name is given highest priority
Lord Krishna may be considered as Purushottama, But Lord Rama was a Maryada Purushottama, means the best person living in community. one step above lord Krishna. Both are the one, but difference between examples provided by Lord Vishnu. It is said in Ramayana that just one name of Rama effects 10 times taking name of Vishnu.
There is also a trust level which is highest for Lord Rama, In india we say
leave it on Ram Bharose, means leave it on the trust on Rama.
People also seek to have a son like Rama, non other any deity. Rama only.
People give way to troupe carrying dead body by just hearing Rama nama Satya he. And also chant the name of Rama and wish peace for the dead person's soul.
This name is holiest word and its chanting creates holy vibrations for soul.