Learn Italian: Non me ne sono accorto, perché leggevo il giornale. - I did not notice it, because I was reading the newspaper.

The verb here is the pronominal verb accorgersene (to notice or realize something), which combines the reflexive verb accorgersi with the pronoun ne. Whenever a reflexive pronoun like mi, ti, ci, or vi is placed right before ne, its ending changes from i to e. Therefore, mi + ne becomes me ne.

What exactly is the ne replacing in this sentence?

It replaces the preposition di plus whatever was supposed to be noticed. The base verb is accorgersi di (to notice [something]). When you don't explicitly name the thing you noticed, you replace the "di + [thing]" with ne, which translates roughly to "of it" or "about it."

Why does this verb use sono instead of ho, and would the ending of accorto ever change?

All reflexive and pronominal verbs (like accorgersene) must use essere as their auxiliary verb in compound tenses. Because it uses essere, the past participle must agree with the subject's gender and number. If a female speaker were saying this sentence, she would say Non me ne sono accorta.