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What is the purpose of のは in the sentence “今浮かんだのは”?
The context is, I asked someone what their favorite English word was and they said “Hmm 今浮かんだのは bullshit. Just kidding!”

Would the translation be “I just came up with ‘bullshit’”? I don’t understand the grammar used here. Please help, thanks!

You: "What's your favorite English word?"

Guy: “Hmm 今[浮]うかんだのは 'bullshit'. Just kidding!”

「の」 here means "(the) one", which of course refers to his favorite English word.

「は」 is the regular topic marker.

By attaching a 「の」 to 「浮かんだ」, 「浮かんだの」 can now be treated just like a noun -- "the one that just came to mind". That is why you can attach the topic marker 「は」 directly to 「浮かんだの」. You following me?

Would the translation be “I just came up with ‘bullshit’”?

If you want it to be, yes, but that is not what it means literally. The grammatical subject of the original is "今浮かんだの" and not an unmentioned first-person pronoun. You used "I" in your TL.

Once again, the original reply means:

"The one that just came to mind is 'bullshit'."

の is a nominalizer and は is the topic particle. To put it simply, a nominalizer turns a non-noun (in this case a verb) into a noun. In English there's a number of ways to do this, but the most common is adding -ing to the verb. For example you say Running is hard not Run is hard. "That which", "The one (who)", etc are other ways to do this in English.

The verb that is being nominalized to act as a noun is 浮かんだ, which is the past tense (た-form) of 浮かぶ. 浮かぶ in this context means "comes to mind".

AのはBだ

is a known as a cleft sentence and it's a pattern that is frequently used in Japanese. It basically introduces A and then tells us A is B. In this case, we have

今浮かんだのは bullshit.

We can clearly see A is 今浮かんだ as a noun: "that which just came to mind". And B is "bullshit". Thus your friend is saying that

that which came to mind just now was "bullshit"

in response to your question of what their favorite English word is. In more natural English ordering you might say

"bullshit" was what came to mind just now.

This is a great answer with more information about cleft sentences

• I don't understand what のも means in this sentence

Also, I think it's good to point out that のは always needs a context. For example, if, out of the blue, I said:

今朝食べたのはパンです。 "The thing I ate this morning was bread."

That would be a little strange. The grammar is perfect, but it is a bit unnatural. The reason is because の　is a stand-in for some kind of noun.

If we have a context, using のは is perfect and natural.

A: 最近食べているものは何ですか？ "What things are you eating recently?"
B: ええと、今朝食べたのはパンです。"Umm... the thing I ate this morning was bread."

Person A uses the noun もの, and in the reply, person B uses the particle の to mean もの.

In a nutshell, のは is most natural in a Q and A situation, when you want to reference the noun that the asker used in your reply:

What animals do you see out the window?
The one I just saw was a bird.