As the title suggests that we want to adjust the size of the labels and not the tick marks I figured that I actually might add something to the question, you need to use the mtext() if you want to specify one of the label sizes, or you can just use par(cex.lab=2) as a simple alternative. Here's a more advanced mtext() example:
set.seed(123)
foo <- data.frame(X = rnorm(10), Y = rnorm(10))
plot(Y ~ X, data=foo,
yaxt="n", ylab="",
xlab="Regular boring x",
pch=16,
col="darkblue")
axis(2,cex.axis=1.2)
mtext("Awesome Y variable", side=2, line=2.2, cex=2)
You may need to adjust the line= option to get the optimal positioning of the text but apart from that it's really easy to use.