The 1GB or 2GB is a measure of dedicated memory, not speed. The GDDR5 would be more an indication of speed (faster than GDDR3).
What is the model number of the Radeon you're looking at?
Current recommended memory for adventure games is more like 256 MB than 1GB, so I'd guess 1GB would be sufficient. But that's a guess. I'd be more concerned with the speed and driver compatibility of the card than with 2 GB over 1 GB.
In general, I'd go with higher specs on a laptop video card since they can't be upgraded (except in some very rare cases).