If so, was there any specific reason behind choosing that name?
Yes, Shri Rama was named by Sage Vasistha:
Valmiki Ramayana Bal kanda
अतीत्य एकादश आहम् तु नाम कर्म तथा अकरोत् | ज्येष्ठम् रामम् महात्मानम् भरतम् कैकयी सुतम् ||१-१८-२१
सौमित्रिम् लक्ष्मणम् इति शत्रुघ्नम् अपरम् तथा | वसिष्ठः परम प्रीतो नामानि कुरुते तदा || १-१८-२२
Elapsed are eleven days and the naming ceremony is performed, then Vashishta, the chief priest, named the high-souled elder one as Rama, Kaikeyi's son as Bharata, and one son of Sumitra as Lakshmana and the other as Shatrughna [1-18-21, 22]
The word 'Rama' is defined as
ramante sarve janaaH guNaiH asmin iti raamaH ' in whom all the people take delight for his virtuousness, thus he is Rama.'
Hanuman in Ram Rahasya Upanishad tells that in the word "राम" . "रा" is taken from Ashtakshari mantra " ॐ नमो नारायणाय " and "म" is taken from Panchakshari mantra "नमः शिवाय" ।
The true meaning of the king among the Rama mantras, as well as of the eight-lettered Narayana and five-lettered Shiva mantras, is spoken of as thus:
Where yogi-s revel in the 2 syllable mantra viz. ‘Ra ma’ where the ra syllable denotes fire
wherein abides illumination. Its nature as Existence-Consciousness-Bliss is regarded as the supreme meaning, the
consonant denoting the unchanging Brahman, and the syllable the energy of
Know that Consonants joined to Syllables are used in breathing; the sound ‘r’, of the
nature of light, is therefore used in action.
The sound ‘ma’, is known as ‘maya’ indicating prosperity; being itself the seed-letter
(bijakshara) is also equal to Brahman itself.