What does the の particle do in relation to verbs?

Question

More specifically, what is the difference between 「見るの」 and 「見たの」 in this context:

時々、子供のころの夢を（x）があります。

How does the の particle function? Is its meaning modified by the 「が」 particle as well?

Answer 1

How does the の particle function?

As mentioned by @Pteromys, the の particle can act as a verb nominalizer. A nominalizer is:

1. (linguistics) Anything, usually an affixed morpheme or a particle, that changes another part of speech into a noun. (Wiktionary)

Normally, の is used in conjunction with the topic marker. It turns the previous sentence into a noun - e.g.:

• チーズを食べる - I eat cheese
• チーズを食べるのは、僕です - I am the one who eats cheese

An even simpler example would be:

• 食べるのは僕です - I am the one who eats

Is its meaning modified by the 「が」 particle as well?

Not really. は and が act the same way as they always do when appended to a nominalized verb, which is where the real meaning comes from. I'll spare you the difference here since there are many resources with those details!

You'll often see のが used in set phrases such as ～のが好きです. Remember though that the の just turns the previous sentence into a noun, so similar to above:

• チーズを食べる - I eat cheese
• チーズを食べるのが好きです - I like to eat cheese

Now before we pull this all together for your first example, first we have to consider whether to use の or the alternative nominalizer こと. Here is a good answer detailing the differences.

TL;DR with の there is often an 'immediacy of time and/or location' whereas with こと, 'matters are considered from a more abstract, removed standpoint'.

• 時々、子供のころの夢を見る - Sometimes I dream about my childhood
• 時々、子供のころの夢を見る「こと」があります - Sometimes I have a dream about my childhood

As for 見た, I don't think it's right either, but I can't really explain thoroughly why. I'll leave that to someone more qualified!