My mother a green card holder went back home for a 6 weeks visit when she came back to the U.S the immigration officer at the airport stamped her passport and wrote ARC 6wks. Does anybody know what ARC means?

I think an ARC stamp is used to show how long a person is out of the country, a record to show mainly for when you go for US Citizenship.

If or when your Mother applies for Citizenship in the US, it will show what length of time she has stayed out of US, I think you can only stay out of the US a certain percentage of time, maybe a senior member can jump in here, as I am not sure if I am correct?

ARC is the acronym for Alien Registration Receipt Card. As far as I know, this code is being used interchangeably with Form I-551 annotation at ports of entry. Not so sure though.

Depika; How far you have reached in your process

I also got that ARC when I entered USA last time.I'm a LPR.

This is required by law to make sure that when you file for your citizenship, you have recored all you visits in form # N-400.
At some stage many applicants have only given three to six months time in all that 5 years which is not correct, as evident from the records, and case was rejected.