"Mis amigos dijeron que te quieren ver mañana."
Translation:My friends said that they want to see you tomorrow.
Is "wanted to see" OK? "Wanted" seems more natural to me but I'm not really sure.
And it is correct. This is the "time shift" of indirect speech. If the initiating verb is in the past tense (here: dijeron - they SAID) then the tense of the original sentence is shifted. Original sentence: Friends: "We want to see you!" - They say they want to see you tomorrow (no time shift) - They said they wanted to see you the next day (time shift; note that you have to change tomorrow into the next day!)
Would it be grammatically correct to say "quieren vérte mañana" in this sentence? As an English speaker it makes more intuitive sense to me, and I think both that and the given sentence are grammatically correct.
I also find the past tense 'said' mixed with present tense 'want' rather than wanted awkward.
In English you could not have "said" followed by "want." You have to say "wanted." You could say: "..they would like to.."
I see nothing wrong with the English translation given: My friends said that they want to see you tomorrow. "Wanted" suggests they have changed their minds and no longer want to see you. "Want" indicates they still want to see you.
Not necessarily. If yesterday they said "We want to see her tomorrow", if you report the speech indirectly, you should shift the verbs in the sentence you're reporting into the past tense. "Yesterday they said they wanted to see you tomorrow" does that and doesn't necessarily imply they've changed their mind. "Yesterday they said they want to see you tomorrow" is used, but it's not really a standard sentence, I think. A similar think is done in Spanish as far as I know and imperfect tense would frequently be used, especially with querer. I'd say "Mis amigos dijeron que te querían ver mañana" should be OK.