Occasional starting problem in 2004 Civic

This is about my son's 2004 Honda Civic LX. On occasion, a turn of the key
won't start the engine. Lights and other stuff work, as they would if key
were in the "accessory" position -- just not the engine. No click of starter
solenoid. After several turns of the key it starts, once in a while as many
as 20, but usually only 3 or 4. Two weeks ago he got a new battery.
And this is odd: It seems to be dependent to how long it has sat. My son
works at his apartment and shops "big," so that he has groceries for two
weeks. If the car has sat that long, it won't start. Taking it in to the shop
is useless because if he tries to start the car soon after pulling in, of
course, it starts.
I am thinking a cheap test would be a starter relay. But the various sites I
have seen have different relay locations. For example,
it, but the sites

't show a starter relay at all! And to add insult to injury, the stupid
owner's manual has only one relay. Sheesh.
Anyway, can someone point me to a good 2004 Civic LX diagram?