I don’t think they mention it. Why do you think it’s strange to say he won the first Nobel Prize? He was one of the people who first were awared one.
Oh, haha that I actually didn’t know, then it makes sense!
Thanks again 
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p. 69
この光の発見は、世界中を驚かせました。
The discovery of this light surprised all around the world.
X線で撮るレントゲン写真は、体の中の病気や、折れた骨などを、調べることができるのです。
Röntgen photographs taken with X-rays made it possible to examine diseases inside the body, broken bones, etc.
レントゲンは、X線の発見によって、最初のノーベル賞を受賞しました。
For the discovery of X-rays, Röntgen won the first Nobel Prize.
でも、X線でお金儲けを 使用 しようとは、思いませんでした。
But he didn’t think that X-rays could be used to earn money that he should make money with X-rays.
EDIT: Thanks to @buburoi for pointing out my mistake on 使用・しよう (see below)
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I think you misinterpreted this one. しよう here is not 使用 but the volitional form of する. So it reads something like: But making money with the X-rays was not something he intended. (Maybe not the best translation but you get the idea.)
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I was a bit surprised because my sentence suggests that he will make money with his discovery. That’s the problem with hiragana instead of proper kanji 
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Haha, yes it’s really annoying sometimes. Today I stumbled upon this sentence: ジャマイカと いえば、元りく上たんきよりせん手の ウサイン・バルトさん… and took me forever to get what it’s about. Kanji would have been helpful, I guess. 
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Here’s my page 68 (again erring on the painfully literal side):
なまりと いっしょに 手の ほねが、けい光ばんに うつったのです！
[Explaining what Röntgen said in the previous sentence:] Together with the lead, the bones of his hand: they were projected onto the fluorescent screen.
『何か』は、ひふや きん肉を 通りぬけ、ほねは 通りぬけなかった ため、ほねの かたちが かげと なって、うつっていたのです。
The “something”—because, though it passed through skin and muscle, it did not pass through bone—became a shadow in the shape of the bones. And so [the bones] were projected.
レントゲンは、この 『何か』を「X線」と 名付け、はっぴょうしました。
Röntgen named the “something” X-rays and made it public.
「X」と いうのは、正体の わからない ものを さす いみで つかわれる 言葉です。
They were called X because that is a term that is used with a meaning that indicates things of which the true form is not understood.
X線は じつは、目には 見えない とくべつな 光だったのです。
X-rays, in actuality, are a special kind of light that is invisible to the eye.
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p．70
「私は、X線が世界の人々のためになるよう、自由に使えるようにしたいのです。」
“I want to make sure that X-rays are freely usable for the benefit of the people of the world.”
X線は、病気や怪我を調べる他にも、工場で作った機械に故障がないかを見つけるのに使われるなど、今もいるいるなところで役立っています。
In addition to examining illnesses and injuries, X-rays can now also be useful for various tasks such as checking for failures in factory-made machines.
ウィルヘルム・コンラート・レントゲン（一八四五〜一九二三年）
Wilhelm Conrad Röntgen (1845-1923)
世界初のレントゲン写真
The world’s first Roentgen photography
レントゲンは、X線を発見したことを明らかにするために、奥さんの手のレントゲン写真を撮影しました。
Röntgen took a Roentgen photography of his wife’s hand to make public his discovery of X-rays.
これが、世界で初めてのレントゲン写真です。
This is the world’s first Roentgen photography.
自分の骨が写った写真を見て、奥さんは気味が悪くなり、「自分の命は、もう少ししか残っていないに違いたい」と思ったそうです。
When his wife saw her own bones reproduced on the photograph, she felt uneasy and thought: “I must have only a little time left to live”.
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I think this is another great translation! The only thing I’d like to add is that my the nuance of そうです in the last sentence got lost in your interpretation.
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Question, page 69: " でも、X線でお金もうけをしようとは、思いませんでした。". What does the “もうけ” mean?
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金儲け is a noun and means money-making.
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This is も + 受け - Jisho.org + をする in volitional form:
“even making money out of them”
EDIT: Apparently not! See the answer above mine, that’s the correct one.
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What does the “ために” mean in the two sentences on page 70, “「私は、X線が世界の人々のためになるよう、自由に使えるようにしたいのです。」” and “レントゲンは、X線を発見したことを明らかにするために、奥さんの手のレントゲン写真を撮影しました。”?
Edit: Is it 為に, for, the reason being, or something?
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Conveniently, sentence 1 goes with meaning 1 and sentence 2 with meaning 2
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Is it? 
I thought this would be 金儲け. Is it ambiguous what they mean here?
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I‘m pretty sure it’s お金儲け 
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Hmm, you’re probably right. It fit the meaning well in my head though 
Sorry for any confusion.
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Don’t worry, confusion is a friend. 
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In case anybody wants to know more about him, there is a museum in the town where he was born (it’s established only a few hundred meters from his birthplace):
Of course now is not the time to think about a museum visit, but they show some pictures of the exhibition and stuff.
For the record: I grew up not too far from that place, and when I was at elementary school, we visited it as a school excursion trip. (And I even returned there as an adult.)
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