Tough to say exactly what happened without being there. However, I agree that after 3 hours of looking, it would be futile to spend further time on this.
It would be unlikely that there would be long-term effects secondary to this - however, as was mentioned in the comments below, an abscess may be possible. Another option would be to perform a CT scan to see if the missing piece can be visualized.
This can be discussed with your personal physician.
Followup with your personal physician is essential.
This answer is not intended as and does not substitute for medical advice - the information presented is for patient education only. Please see your personal physician for further evaluation of your individual case.
It's an unusual story. However, there comes a point in surgery when one has to decide whether carrying on might do more harm than good. It's hard to understand exactly what went on; on the other hand, I'd agree that if it wasn't found after three hours it seems unlikely that further looking would help. Here's the good news: it's not likely to "poison" her simply by being there. There's dead tissue inside after nearly any operation (we tie things off, and what's beyond the tie dies away, for example. When there's appendicitis or other internal infections, we know we leave tissue behind which has little or no blood supply and which harbors bacteria. Yet it heals, because that's what the body is able to do) and the body absorbs it. In the case of her appendix, it'll either slowly absorb and she'll be fine, or -- worst case sceanario -- an abscess might form which would require drainage. In that case, it's most likely it could be done by guiding a tube into the area using xray, without the need for more surgery. I'd assume she'd be kept on antibiotics a bit longer than usual postop. Her body will take care of the rest, one way or another, and it's HIGHLY unlikely that there'd by any long-term after effects.
added comment: a CT scan would be highly unlikely to show it. If and when there's reason to be concerned about an abscess, a CT would be the xray of choice.