I love this game. I think I would send the idea of plant breeding. They had this idea to some extent, but I don't think there was big public, long term effort to breed new crops or get much higher yields. If you get modern crops faster, you have larger populations, which leads to industrialization sooner. This isn't just limited to food crops, it's possible ancients could have bred plants and animals for many useful purposes.
Slightly relevant, what information would you send to the past if you had a phone that could communicate backwards in time? But there's a catch, any idea you send is converted to the closest analogue in ancient times. So if you tell them about the scientific method, they will just think about religion. http://lesswrong.com/lw/h5/archimedess_chronophone/
Regarding Archimedes' chronophone, I am not sure that the premise is well-formed. I find I cannot predict what the chronophone will do because the notion of "closest analogue" is inherently ambiguous. In the worst case we cannot send any information back at all by definition.
All that being said, assuming the chronophone will let me, I'd bring their attention to the idea of the printing press. It's easily made, the benefit is obvious, and the effect is dramatic. Kind of amazing it wasn't invented earlier.
I think it's a self-defeating idea, but a useful thought experiment. It makes you think about the abstract principles behind things that are good ideas, and then the more-abstract principles behind those.
The link explains it further. The point of the chronophone is that you can't just send ideas you already have back. If you want to send the scientific method back in time, you have to actually do science. If you want to send the printing press, you actually have to think about better ways your present civilization could transmit information.
Slightly relevant, what information would you send to the past if you had a phone that could communicate backwards in time? But there's a catch, any idea you send is converted to the closest analogue in ancient times. So if you tell them about the scientific method, they will just think about religion. http://lesswrong.com/lw/h5/archimedess_chronophone/