Which part of a female mantises's DNA causes her to be a few times larger than a male mantis?
Insects and spiders have tremendous sexual dimorphism, with males often being much smaller than females.  They are pretty much just sperm carriers and sometimes do not eat.  Insects may have two or one sex chromosome, where males my have only one chromosome.   As is pointed out in the comments, not all mantis' species display this dimorphism.  

Since such dimorphism is probably due to small changes in the genome due to environmental selection pressures and the genetic mechanisms of control are probably not simple.

Its difficult to say which genes are responsible for size per-se.  Recent studies of human height indicate that scores of genes may be involved in human size and insects have at least as many genes as humans do; it is pretty hard to look at a human genome sequence and say what height that person is without some other help.  I imagine for specific insect species this too would be difficult.