"The letters contain mainly the same annoying idea that he was anxious to see his friends doing experiments outdoors."
Without being able to see what lies before or after your sentence makes advising you a bit difficult but here are a few thoughts that I had.
I think that your discomfort with using the word "that" stems from correct tense, punctuation issues, and or the sentence's murky (unclear), object.
"The letters [contain] mainly the same annoying idea that he was anxious [to see] his friends doing experiments outdoors."
"contain" is present tense / 'to see' is future tense so if we correct that and add punctuation before "that" -
(Written this way, "that" would be correct.)
The letters contained mainly the same annoying idea - that he was anxious to see his friends doing experiments outdoors.
The sentence (the way you have written it), leaves the reader wondering what the intended subject, or focus, is.
The author of the letters?
What is the object of "annoying idea"?
What is the object of his anxiousness?
That his friends were doing the experiments outdoors?
Mainly, all of the letters conveyed the same idea; he was excited that he was going to see his friends doing experiments outdoors.
Mainly, all of the letters conveyed the same idea; that he was anxious about his friends doing the experiments outdoors.
The letters were annoying because they all said pretty much the same thing -
"My friends are going to be doing some experiments outside, I can't wait!"
The letters contain no clues. They are mainly about someone's friends doing experiments.
Hope that helps.