Ne pas vouloir de is something of an emphatic construction that specifically rejects a situation that could befall you for some reason, as here, because the Prince is being offered one.

This is why when you answer in the negative to a question with veux/voulez-vous [+ objet], the formulation is technically “Je n’en veux pas“, where the en implies a phrase with de.

When you’re saying “je ne veux pas” with a following noun phrase, you’re usually implying that you want something else, which you’re going to promptly explain. The de construction specifically rejects the possibility without implying anything further.

It is one of the different possible constructions for the verb vouloir.
(which is indeed one of these overused verbs which have a huge number of meanings and constructions)

Vouloir de [X] means “accepter [X]”, which slightly differs from the main meaning of the verb. It is therefore used only to refer to something after it has been proposed to someone. Example : “Depuis une heure je lui explique que c’est plein de vitamines mais elle n’en veut pas !”

It can also be used to refer to someone rather than something, and sometimes it’s an implied reference to wedding proposal. As in :

« Je te construirai un chateau si tu veux de moi ! »

More on this subject in the TLFi article here (II, A)

There are several meaning nuances for the verb “vouloir”. Here, your case corresponds to this one in Wiktionary :

11. (vouloir de, généralement sous forme négative) Accepter, malgré des conditions ou des conséquences défavorables.

There’s the idea of acceptance and it is mainly use in the case where you refuse a proposal.

Some examples:

• Je veux faire quelque chose = I want to do something

• Je veux un ballon = I want (desir to have) a ballon

• Je ne veux pas un ballon, je veux deux gateaux = I don’t want a ballon, I want two cakes (You are expressing a choice, a decision. In this context, after having said what you don’t want, you often say in addition what you do want. It can be another quantity or something else instead, or both something else and in another quantity)

• Je ne veux pas d’un ballon, garde-le ! = I don’t want a ballon, keep it! (This is used in the case you refuse a proposal. The contracted version of the same sentence would be: Je n’en veux pas, garde-le ! = I don’t want it, keep it!)

PS : Dans ton cas, l’auteur donne au Petit Prince un dessin et ce dernier n’est pas content, il refuse le dessin et dit qu’il n’en veut pas.

I’m 99% sure this means something like “I don’t want an elephant inside a boa”.

Not really. I think it means something more like “I want nothing to do with…“, rather than “I don’t want…“. It’s a rejection of the entire idea, not just an expression of not wanting it. A somewhat more literal translation might be I want nothing of it — the “of it” is like the “d’un“.

Romain & Circeus have answered this well. It’s about vouloir de, not just vouloir.

I’d suggest that equivalent English, for the negation, is something along the lines of your not wanting to have anything to do with it.

“Je ne veux pas d’un elephant dans un boa” means, essentially, “I want nothing to do with any elephant that is inside a boa” (or “I want nothing to do with a boa that has an elephant inside it“). One might even be inclined to add, “I don’t even want to think about it.“