If you want to chose a random duplicate to keep, rather than duplicated
s default behaviour of only keeping the first, then why not randomly shuffle the whole dataset, so that choosing the first in the shuffled set is effectively a random row from the original:
DATAr <- DATA[sample(1:nrow(DATA),]
DATAr <- DATAr[!duplicated(DATAr$Point),]
If the order of your original DATA was inportant, store the sample(...)
in a variable, use that to re-order your data, and apply an inverse once you've removed duplicates (or add a column DATA$ind <- 1:nrow(DATA)
and sort your data to restore this afterwards.