In the sentence "John told James that he's happy.", the pronoun "he" is ambiguous, since it could refer to either John or James.
Are there any languages which try to solve this kind of ambiguity by having different pronouns depending on whether the last, second-to-last, etc. antecedent is referred to?
Yes there are. Examples include Greenlandic and Cree.
It's not exactly what you asked for, as it doesn't depend on whether it's the last antecedent, or second-to-last antecedent. But in these languages, 3rd person pronouns are in two categories; proximate and obviative (obviative is sometimes called fourth person). The proximate one is one that the conversation actually is about, and the obviative is some other third person. Like this:
I saw John the other day. He looked pretty messed up. He had gotten drunk the day before and gotten into a fight with Dave. He punched him so hard that he broke his rib.
Now English is ambiguous about who punched whom, who broke whose rib. But in Greenlandic, because the discourse is about John, and I'm telling you about that time I saw him looking messed up, John is the salient one, and so Dave is the obviative one, so the pronouns will be selected to match. I don't know Greenlandic enough to translate your example. But I found a similar example on the Wikipedia article on Greenlandic language:
Ole oqar-poq tillu-kkiga "Ole said I had hit him (the other man)"
Ole say-3p hit-I/3p
Ole oqar-poq tillu-kkini "Ole said I had hit him (Ole)"
Ole say-3p hit-I/4p
Here the verb ending may be -kkini or -kkiga. These verb endings mark the clause as subordinate, they agree with the verb's agent and with the verb's patient. The difference is that -kkini needs to be coreferent with someone or something in the matrix clause or in earlier discourse, but -kkiga can't be.
Some languages (such as Potawatomi) even have a further obviative, a grammatical person which is one step removed again from the discourse. It could have been useful if in that story, Dave had a dog with him for example.
In some sign languages, pointing is used as a pronoun. It makes different distinctions to the ones made by English pronouns.
In English, he, she, this, that and it are different. He and him are different. These distinctions are not made in the sign languages of English speaking countries.
Instead, you can point directly at a person or thing instead of using its name. That provides as many distinct pronouns as there are things or people that are mutually visible to the speaker and audience.
If Alice points at Bob, then Bob would translate that into English as "you" and Charlie would translate it as "him" but unlike an English "you" or "him", everyone who saw it would unambiguously know that it is about Bob and Bob only.
In ASL, BSL and Auslan, pointing with the index finger is I/you/he/she/that and so on, and there's a way to point with the palm for the possessive pronoun.
In a sign language you may also be able to point at a location in empty space as a pronoun that means "the person or thing I talked about after I pointed here" or "the person or thing I talked about while my hands were over here". This provides a few pronouns that can distinguish in context a small number of different subjects that aren't mutually visible.
For "John hates James, he's rude" in Auslan Alice could sign "right James, left John hates, he's rude" and she could indicate whether James or John is the rude one by signing the "he" to the left or right. ASL might have different word order to this example (I don't know) but a similar principle would apply.
This link gives specific information about pronouns in ASL, with videos. https://www.handspeak.com/learn/index.php?id=27
BSL and Auslan are very very different languages to ASL but some of the pronouns are the same in all three.
Aside from obviative third person pronouns mentioned by OmarL, some languages have what are known as 'reflexive' pronouns. These pronouns refer directly back to the subject of the clause that they are part of (or the parent clause if they are the subject of their clause), and thus can either partially (if their use is optional) or completely (if they are mandatory to express the sentence form) resolve this ambiguity.
Swedish, for example, has specific reflexive third-person pronouns (in the cases of verb phrases that require reflexive pronouns, the objective forms are used instead for first and second person pronouns), and their usage is functionally required for expressing this relation. 'sig' is used when the pronoun is an object in it's clause, and 'sin', 'sitt', and 'sina' are used for reflexive possessives (Swedish possessives have to agree in gender and plurality with the noun being possessed).
In Swedish for example, the English sentence 'Jakob ate his food.' could be translated as either of:
The first one states that James ate someone's food and implies that it is not Jakob's food that Jakob is eating, while the second one specifically states that Jakob is eating his own food.
One feature that disambiguates your specific example is logophoricity.
A logophoric pronoun refers to the speaker/thinker/writer/feeler in the higher clause.
Ewe is a Niger-Congo language with this feature, example taken from the Wikipedia article.
a. Kofi be yè-dzo
Kofi say LOG-leave
Kofiₐ said that heₐ left.
b. Kofi be e-dzo
Kofi say pro-leave
Kofiₐ said that he/sheₒ left.
There are some restrictions on the use of this construction in Ewe: for instance, the antecedent must be a subject in the clause it appears in.
Some languages feature logophoric marking on the verb itself.
This example is from Akɔɔse, a Bantu language spoken in Nigeria.
a. à-hɔbé ǎ á-kàg
he-said RP he-should.go
Heₐ said that heₒ should go
b. à-hɔbé ǎ mə-kàg
he-said RP LOG-should.go
Heₐ said that heₐ should go
There are also languages that are differently ambiguous in this context because the meaning of the first and second person shifts in certain embedded clauses.
From this article by Pranav Anand & Andrew Nevins, we have an example where the first person singular pronoun εz shifts inside embedded clauses associated with certain verbs.
The examples are from Zazaki, an Indo-European language from eastern Turkey.
Hεseni (mɨk-ra) va kε εz dεwletia
Hesen.OBL (I.OBL-tO) said that I rich.be-PRES
Hesen said that {I am, Hesen is} rich
This is not the exact answer to the question, but I think it is still relevant. The Czech language still has this particular ambiguity the same as English in this particular sentence, but I think that since you asked about sentence ambiguity it is relevant of me to point out that Czech sentences are unambiguous when speaking in similar certain contexts.
An example is "John was there with his girlfriend" - it is ambiguous if the girlfriend is actually John's or some other man's. In Czech it is one of either:
John tam byl se svou přítelkyní.
John there was with his girlfriend.
John tam byl s jeho přítelkyní.
The first sentence with the word "svou" says the girlfriend is John's, while the second sentence with "jeho" means the girlfriend was some other man's. If it was some other woman's girlfriend, the word would be "její".
Please go easy on me if I did something wrong or explained myself poorly while answering, I'm by no means a linguist, I'm just a Czech guy who found this question interesting.
It's a constructed language, but Lojban has a number of ways to resolve this, which is not surprising since one of its main design goals is exactly to allow speakers to avoid ambiguity.
The first one is very similar to what you propose. In Lojban the "pronoun" ri is used to refer to the last "noun" (Lojban doesn't have the same notion of nouns and pronouns as English) mentioned. The similar pronouns ra and ru have a less precise meaning, but are supposed to refer to progressively more distant nouns. So in Lojban one can say the following:
la djan. jungau la djeimz. lo du'u ri gleki
John told James that he (James) is happy.
la djan. jungau la djeimz. lo du'u ra gleki
John told James that he (John) is happy.
If that's not precise enough, Lojban also has assignable pronouns. So if you know that you will be referring to something again later, you can assign it an identifier in the middle of the sentence:
la djan. jungau la djeimz. goi ko'a lo du'u ko'a gleki
John told James (= ko'a) that "ko'a" (i.e. James) is happy.
la djan. goi ko'a jungau la djeimz. lo du'u ko'a gleki
John (= ko'a) told James that "ko'a" (i.e. John) is happy.
There are a total of 10 of these assignable pronouns: ko'a, ko'e, ko'i, ko'o, ko'u, fo'a, fo'e, fo'i, fo'o, fo'u, so you can have multiple different nouns assigned at the same time. This is more useful for writing though, and can get cumbersome in speech.
There are still other ways to resolve this kind of ambiguity, for example there are ways in Lojban to explicitly refer to the subject or a specific object of the current or previous sentence. For those interested in going down this rabbit hole I recommend checking out the Lojban Reference Grammar.
Arabic has a grammatical way of addressing this, not by using different pronouns, but by adding diacritics to the adjective to match either the subject or the object, such as::
قالَ جونٌ لجيمسً انهُ سعيدٌ
قالَ جونٌ لجيمسً انهُ سعيدً
In the first sentence the diacritics on the adjective "happy" matches the subject "John", and in the second it matches the object "James".
In Polish this sentence is not ambiguous.
The most direct, word-by-word Polish translation of this sentence is: "John powiedział Jamesowi że jest szczęśliwy" - and for normal user of Polish language this sentence unambiguously mean that it is John who is happy. If we meant that James is happy, we would say, for instance, "John powiedział Jamesowi że James jest szczęśliwy" (direct translation: "John told James that James is happy") or maybe "John powiedział Jamesowi że ten jest szczęśliwy".
