From TCGA barcode I see that plate and center are different but the sample ID (TCGA-A6-2684-01) is same. Which one should I prefer for the analysis? Do I need to keep both the samples? When I consider sample ID it will be duplicate samples.
In the example that you have given, they are indeed the same sample and are just different aliquots from solid normal tissue:
To get that screenshot, I went to the GDC and put one of the sample names into the search box.
It is for cases like this that I wished that the maintainers of these TCGA R packages did more curation of their data, but I admit that it is a lot of data and that, as researchers, we may not even have funding left for such things.
With regard to which one you choose, you could just obtain the mean value of both, or literally just discard 1. Either way, to report this in the methods is a single line which will be brushed over by reviewers.
Just to add a quick note to Kevin's helpful answer-because there can be various duplicated samples, in any conditions, as there are also various guidelines that you could take into account, from the BROAD institute, like the following example from the READ dataset: