you can use the left_on and right_on options as follows:
pd.merge(frame_1, frame_2, left_on='county_ID', right_on='countyid')
I was not sure from the question if you only wanted to merge if the key was in the left hand dataframe. If that is the case then the following will do that (the above will in effect do a many to many merge)
pd.merge(frame_1, frame_2, how='left', left_on='county_ID', right_on='countyid')