Why is “schreiben” capitalized here?
Ich brauche etwas zu schreiben = I need something to write
Ich brauche etwas zum Schreiben = I need something to write with (literally “I need something to the writing”, which doesn’t work in English)
The capitalization changes the way we’re using the word “schreiben” (from a verb to a noun), and so changes the meaning of the sentence.
I think “Ich brauche etwas, womit ich schreiben kann” more closely matches the suggested English translation, but I don’t know which would sound more natural to a German native speaker.
For me, as a native speaker, „Ich brauche etwas zum Schreiben.“ sounds more natural.
That is not so easy.
“ich brauche etwas, womit ich schreiben kann,” is quite formal und it definitely asks for a tool for writing, e.g. a piece of paper and a pen. As you say in English that would be “I need something to write with.”
“Ich brauche etwas zum Schreiben” is colloquial. It asks for something to enable the writing, but not necessarily a tool. It might be a pen but it might also be an alcoholic writer asking for a bottle.
A less colloquial sentence with the same broad meaning would be “Ich brauche etwas, um zu schreiben.”
English Translation
I need something to write with.
“Ich brauche etwas zum Schreiben”
“Ich brauche etwas, um zu schreiben.”
Can this “etwas” in these two sentences be an idea or a story, namely the content to write (about)?
No, that would be:
“Ich brauche etwas, um darüber zu schreiben.”