I want to talk about you
Nanji is a Japanese pronoun (yeah, they totally exist) that means what "thou" does in modern English: "you [+archaic]". If you call someone nanji, you are speaking in the voice of either an ancient religious text, a venerable Chinese poet, or a long-dead Japanese nobleman*.
The na- part is an OJ second-person demonstrative, as seen in the Man'yoshu:
石室戸尓 立在松樹 汝乎見者 昔人乎 相見如之
ihayato ni/ tateru matu no ki/ na wo mireba/ mukasi no hito wo/ ahimiru gotosi
Pine that stands by the mouth of the cave
I look at you; it is like coming face-to-face
with somebody from long ago
(Fun fact: Author Hakutsū Hōshi is talking about the Mio caves in Wakayama.)
And of course it had a -re form like kore, ware, etc:
朝井代尓 来鳴杲鳥 汝谷文 君丹戀八 時不終鳴
asa wide ni/ ki naku kahodori/ nare dani mo/ kimi ni kohure ya/ toki ohezu naku
Kaho-birds that come and cry at the dam at dawn
Can you be in love with them too?
Ceaseless in your cries
Note use of kimi to mean "my lover" in a third-person rather than second-person way. Also note that the kaho in kaho-tori might be onomatopoeic (which might in turn make it the cuckoo).
The -nji comes from muti (that is, nanji was originally namuti), which Bjarke Frellesvig explains as meaning "'esteemed person; honorific suffix in names and titles', cf. mutu- hon. prefix)." Ōno agrees, citing sumemutsu kamuroki as an example. (I think that particular phrase was used to for the emperor's grandfather.) This is the mutu- that survives in modern Japanese as mutsumajii, "harmonious".
Unfortunately there aren't any good examples of namuti in the Man'yoshu—the sound appears a few times, but only in ohonamuti, which was another name for Ōkuninushi and elsewhere appears in enough variant forms like ohonamoti ("possessor of the great name") and ohoanamoti ("possessor of the great hole") to make any direct relationship with namuti seem unlikely.
Simon:
The Japanese Lord's Prayer still ends 限りなく汝のものなればなり。"Thou" is not merely an archaic form of "you", however; it's a 2nd person singular where "ye" was 2nd person plural, a hangover from our German inheritance. (And, of course "Ye Olde..." is a problem with orthography, not grammar.)
Incidentally, Bjarke Frellesvig used to be my tutor. Which, scarily, suggests that I used to know all this at some point in the past. Lost in time, like tears in the rain.