A few ways to do this, the simplest might be to call geom_point twice, subsetting based on that column value.
library(ggplot2)
set.seed(12341234)
dataframe <- data.frame(col1 = rnorm(10, mean = 2, sd = 3),
                        col2 = rnorm(10, mean = 4, sd = 2),
                        col3 = c(rep("A", 4), rep("B", 4), NA, NA))
ggplot(dataframe[!is.na(dataframe$col3),], 
       aes(x = col1, 
           y = col2)) +  
  geom_point(aes(color = col3), 
             size = 4) + 
  geom_point(data = dataframe[is.na(dataframe$col3),], 
             color = "blue", 
             size = 8) + 
  scale_color_manual(breaks = c("A", "B"), 
                     values = c("black", "red"))

Note this will mess up the legend (if you need one).
Another option is to replace NA values with something like "NA".