What is the meaning of "half bit" in this context?
In both of the cases you mention, an analog quantity is involved somewhere. 

In the RS-232 case, the number of “stop bits” refers to a duration, so 1.5 stop bits simply means 1.5 times the duration of a bit at the given rate.

In the ADC case, 0.5bit precision refers to the analog quantity being converted, so 0.5bits essentially means that the digital result is guaranteed to represent the analog quantity rounded correctly to the nearest possible digital representation.