The story of Narada asking the above-mentioned boon from Lord Rama is described in Chapter 126 (pg 723) of the Aranya-Kanda of Ramacharitmanas. Sanskrit text taken from here.
Narada goes to meet Lord Rama because he feels that he may not get the chance to meet him again.
मोर साप करि अंगीकारा। सहत राम नाना दुख भारा ॥ ऐसे प्रभुहि बिलोकउँ
जाई। पुनि न बनिहि अस अवसरु आई ॥
mora såpa kari amgikårå, sahata råma nånå dukha bhårå.3. aise prabhuhi
bilokau° jåi, puni na banihi asa avasaru åi.
"It is in submission to my curse that the Lord is undergoing many
hardships of an oppressive nature. Let me, therefore, go and see such
a noble Lord; for such an opportunity may not present itself again"
यह बिचारि नारद कर बीना। गए जहाँ प्रभु सुख आसीना ॥ गावत राम चरित मृदु
बानी। प्रेम सहित बहु भाँति बखानी ॥
yaha bicåri nårada kara binå, gae jahån prabhu sukha åsinå.4. gåvata
råma carita mrdu båni, prema sahita bahu bhåti bakhåni.
Reflecting thus Nårada went, lute in hand, to the spot where the Lord
was sitting at ease. He fondly sang in a soft voice the exploits of
Lord Råma dwelling upon them in all detail.
After this welcome, Narada asks Rama for the boon that His name surpass all other names in their potency and destroy thousands of sins.
दो. नाना बिधि बिनती करि प्रभु प्रसन्न जियँ जानि।
नारद बोले बचन तब जोरि सरोरुह पानि ॥ ४१ ॥
nånå bidhi binati kari prabhu prasanna jiya jåni, nårada bole bacana
taba jori saroruha påni.41
After much supplication and realizing that the Lord was pleased at
heart, Nårada joined his lotus palms and spoke as follows:-
सुनहु उदार सहज रघुनायक। सुंदर अगम सुगम बर दायक ॥ देहु एक बर मागउँ
स्वामी। जद्यपि जानत अंतरजामी ॥
sunahu udåra sahaja raghunåyaka, sundara agama sugama bara dåyaka.
dehu eka bara mågau° svåmi, jadyapi jånata antarajåmi.1
Listen, O Lord of Raghus, generous by nature as You are: You confer
delightful boons that are unattainable as well as those that are
attainable. Grant me, my master, only one boon that I ask of You,
even though You already know it (even without my asking), indwelling
as You do the hearts of all.
जानहु मुनि तुम्ह मोर सुभाऊ। जन सन कबहुँ कि करउँ दुराऊ ॥ कवन बस्तु असि
प्रिय मोहि लागी। जो मुनिबर न सकहु तुम्ह मागी ॥
jånahu muni tumha mora subhåu, jana sana kabahu ki karau duråµu.
kavana bastu asi priya mohi lågi, jo munibara na sakahu tumha mågi.2.
You know my disposition, dear sage: do I ever hide anything from my
devotees? What object do I hold so dear, O chief of sages, that you
may not ask it of Me ?
जन कहुँ कछु अदेय नहिं मोरें। अस बिस्वास तजहु जनि भोरें ॥ तब नारद बोले
हरषाई । अस बर मागउँ करउँ ढिठाई ॥
jana kahu kachu adeya nahi more, asa bisvåsa tajahu jani bhore. taba
nårada bole harasåi, asa bara mågau karau dhithai.3.
There is nothing which I may withhold from my votary: never give up
this belief even by mistake.
Then Nårada gladly said, This is the boon I presume to ask:
जद्यपि प्रभु के नाम अनेका। श्रुति कह अधिक एक तें एका ॥ राम सकल नामन्ह
ते अधिका। होउ नाथ अघ खग गन बधिका ॥
jadyapi prabhu ke nåma anekå, sruti kaha adhika eka te ekå. råma
sakala nåmanha te adhikå, hou nåtha agha khaga gana badhikå.4.
Even though my lord has many names, each greater than the rest, as the
Vedas declare, let the name RÅMA, my lord, surpass all other names in
exterminating the whole brood of sins even as a fowler kills an entire
flock of birds.
दो. राका रजनी भगति तव राम नाम सोइ सोम।
अपर नाम उडगन बिमल बसुहुँ भगत उर ब्योम ॥ ४२(क) ॥
råkå rajan∂ bhagati tava råma nåma soi soma, apara nåma uŒagana bimala
basahu bhagata ura byoma. 42(A)
May the name RÅMA shine as the moon and the other names as so many
stars in the cloudless sky of Your devotees heart during the full-moon
night of devotion to You.
एवमस्तु मुनि सन कहेउ कृपासिंधु रघुनाथ। तब नारद मन हरष अति प्रभु पद
नायउ माथ ॥ ४२(ख) ॥
evamastu muni sana kaheu krpåsindhu raghunåtha, taba nårada mana
harasa ati prabhu pada nåyau måtha. 42(B)
The all-merciful Lord of Raghus replied to the sage, "So be it"
Thereupon Nårada felt much delighted at heart and bowed at the Lord's
feet.