The story of the name, from the sources
In Lisan al-Arab, al-reem is the white gazelle, pure white; its older form has a hamza: ri-im.
“the white gazelle, pure white”
رِئْم ← رِيم
Lisan al-Arab · vol. 12, pp. 259–261

For more than fourteen centuries, the reem has been the Arab poets' measure of beauty. Al-A'sha said of his beloved: she has the eyes of a reem. Jamil said: the eyes of a reem, and the neck of a fawn.
«لها مُقلتا ريمٍ»
al-A’sha: “she has the eyes of a reem”
«لها مُقلتا ريمٍ وجِيدُ جَدايةٍ»
Jamil: “the eyes of a reem, the neck of a fawn”
al-Aghani · vol. 9, p. 125 · vol. 8, p. 125

Reem is also a valley, thirty miles from Medina: Batn Reem. The Companion Abdullah ibn Buhayna lived there, and fasted all year. Ibn Hajar spells it with the hamza: Batn Ri-im.
بطنُ ريم
Batn Reem, a valley thirty miles from Medina
home of the Companion Abdullah ibn Buhayna
Ibn Sa’d, al-Tabaqat · vol. 4, p. 255 · Ibn Hajar, al-Isaba · vol. 4, p. 190

In Baghdad, twelve hundred years ago, the poet Ashja al-Sulami loved a slave-girl named Reem, and called her name in his elegy for Harun al-Rashid: O Reem.
Baghdad, twelve hundred years ago
«قليلٌ وربِّ البيتِ يا ريمُ ما أرى…»
Ashja’ al-Sulami, elegy for Harun al-Rashid
al-Aghani · vol. 18, pp. 243–244

So does a name shape its bearer? Ibn al-Qayyim believed names are moulds for meanings. The Prophet, peace be upon him, asked a man his name. Hazn, he said: rough ground. You are Sahl: smooth. The man refused. His grandson, Sa'id ibn al-Musayyib, said: the roughness has stayed in our family ever since.
“Names are moulds for meanings” · Ibn al-Qayyim
حَزْن ← سَهْل
“The roughness has stayed in us ever since.”
Sahih al-Bukhari 6190 · Zad al-Ma’ad 2/307
And modern science? A 2017 study found that strangers match a name to its face more often than chance. But no study shows that a name decides who you are.
Reem · ريم · Rim · Rym · ريم · Rîm
2017 study: strangers match names to faces better than chance
No study shows a name decides who you are
Zwebner et al., JPSP 112(4), 2017 · doi 10.1037/pspa0000076
