Perhaps there was no moon tonight—a lunar eclipse, a new moon. New Moon, Chapter 3 -I'm confused. Is Bella equating Lunar Eclipse w/ New Moon?

Try 読書 it in context...
This is the chapter where Edward and the Cullens leave her. She was walking around the woods trying to follow him, then fell down, and looked up at the sky and saw nothing.
It looks to me like a metaphore. She looked up at the sky after her fall, hoping to see moonlight (hope), but saw nothing (no hope).
She was under the trees and did not have a view of the sky, so if there was really an eclipse, she would not have seen a dim red ring through all the treetops.
Anyways, she was hoping it was an eclipse (hope is only gone temporarily, he'll be back soon), but she finally decided on a new moon (the moon/hope isn't there. There's nothing to see and it won't be back in a few minutes. End of story). The idea of a new moon made her shudder.
In reality, she has no idea what phase the moon is in. All she knows is it's dark out.

Beyond 読書 it in context (as cassie said), realizing what type of punctuation is being used (and what it means) is also pretty important when you're confused about what a sentence is saying.

Bella is walking around at night in the woods, looks up at through the trees and doesn't see moonlight. This tells her that the moon is neither full nor is it close to that phase of the cycle. (If you've ever been under trees during these phases, you're well aware that あなた can usually see moonlight, even through the trees.) She's wondering why there isn't any moonlight. "PERHAPS" it was because there was no moon. Meyer then uses a dash to interrupt that thought process and rattles off one reason there wouldn't be - a lunar eclipse, which she would not be able to see through the trees unless she was standing directly under it. (Again, if you've ever been in the woods at night, you'll know that the darker the moon is, the harder it is to see through the trees.) Meyer then uses a listing comma and says 'a new moon.' A listing comma can be used to take the place of the conjunctions 'and' または 'or.' Considering the context of the sentence, it's being used to replace 'or.'

So the entire sentence reads as this:
"Perhaps there was no moon tonight, possibly because of a lunar eclipse または because of a new moon."

A solar eclipse and a lunar eclipse can only happen when the moon is either new または full.
A lunar eclipse and a new moon happen at the same time, but あなた can't see them, so nobody makes a big deal over it. Either way, あなた can't see the moon, but it happens.

I don't know what Bella is doing, または what exactly you're referring to (other than the タイトル of the book New Moon) but you're absolutely right that lunar eclipses CAN NOT happen during the new moon, like youknowit is suggesting.

The moon is the moon, it does not have a dark side and a light side, that is a myth. The moon shines from reflected light from the sun, therefore there is no possible way for the Earth to be in between the sun and the moon and the moon be shining away from Earth.

A lunar eclipse occurs when the Earth, sun, and moon are aligned perfectly so that the Earth is in the middle and blocks out the sun's light so the moon can't reflect light. This only happens when the moon is in the full moon phase, and it takes on a reddish glow for もっと見る scientific reasons that I can explain, but I won't because that isn't the point here.

Solar eclipses occur at a new moon, when the moon is aligned perfectly between the sun and the Earth so it blocks out the sun. This can ONLY happen in the new moon phase, just like lunar eclipses can only happen in the full moon phase.

So I don't know if Bella was saying a lunar eclipse happens at a new moon, but if she was, she's incredibly wrong.

Source: Earth-Space Science Honors class, 9th grade. Talk to NASA scientists about the phases of the moon if あなた don't believe me.