IMO a common solution here is
"She doesn't eat meat. I think she's a ..." he trailed off.
"Vegetarian?" I continued for her.
or
"She doesn't eat meat. I think she's a ..." he trailed off.
"Vegetarian?" I finished the sentence.
Note that for me, "he trailed off" doesn't really work like that.
or
"She doesn't eat meat. I think she's a ..." he said, trailing off.
"... vegetarian?" I picked up.
Just to be clear, "volunteered" (see other answer) is perfect here, if that's the feel you want. (If you want "brusquely added" ... say that. If you want "quickly completed" ... say that. if you want "hesitatingly suggested" ... say that.)
{Don't forget ... writing is about actions. The words are nothing, zero. It's like drawing is about looking. The pencil marks are nothing, zero.}