The standard meaning for this word seems to be “well”, sometimes positive, sometimes negative, sometimes simply as a placeholder (as was also explained in answers here).
But this seems to be only the case when it is at the start of a sentence. However, I have also seen it used a lot as an interjection on its own with an exclamation point (“Naja!”).
Does this change the meaning of the word? What is one trying to say by simply responding “Naja!”?
The meaning of a single "naja" would depend on its context. Prosody might be a stronger indicator than the actual word itself. You won't find it much in written language, too, apart from direct speech maybe, or colloquial texts.
Basically, I'd describe it as some kind of "meh"-like utterance (or "well", as you said), signaling (implied) lack of importance/relevance of something said, for example to change topic, or to save someone's face while still criticizing.
A: ... und deswegen sind wir dann im Kino gelandet. Naja! Später hat Heiner ...
(A: ... and that's why we ended up in cinema. Well! Later, Heiner has ...)
A: Papa fände es bestimmt auch schöner, wenn hier nicht so ein Chaos wäre. Naja! Essen ist fertig.
(A: Dad would surely prefer, too, if it wasn't such a mess in here. Oh well! Dinner's ready.)
But it can also have a dimension of not-really-agreeing. Then it might even appear as a doubled "naja", functioning mainly as a turn taking instrument introducing an objection.
A: Jedenfalls sind wir zu dem Schluss gekommen, dass das so schon richtig war.
B: Naja, naja! Also, ich denke ...
(A: Anyway, we came to the conclusion that it was the right thing to do.
B: Wait a minute, actually, I think ...)
The "naja" signals disagreement which is explained in the following utterance. The disagreement does not always have to be further explained, though that might leave a communicative gap which the other participant(s) will want to close.
A: Mamas Spätzle sind einfach die besten.
B: Naja.
A: Wie meinst du das?
(A: Mom's spaetzle still are the best.
B: Meh.
A: What do you mean?)
Sorry for the rough translations, my English is probably not good enough to convey the exact same meaning. All in all, it's a highly contextual kind of word which predominates in oral communication and thus is difficult to pin point, especially if you don't know how it is pronounced. Still, it's an everday expression flowing in many conversations, so if you are actually talking to someone, it might not be that much of a problem after all.
What is one trying to say by simply responding “Naja!”?
When "simply responding Naja", that can be for closure of a (possibly but not necessarily unpleasant) topic. You'd lower your tone here.
Naja.
as in
Let's leave it like that
It can be used to announce an objection or a further thought, too. In this case, however, the intonation is rising and it must follow something (objection/thought)
Naja! [..]
