Have same problem, but WORSE. Fails to send signal to monitor BOTH on new boot, AND on trying to wake from sleep (both situations about 75% of the time). MacMini Oct 2014 with OS X 10.2 From day of purchase, no signal to monitor using DisplayPort to DVI on HP Monitor. Checked settings on monitor, no change. Moved mouse around. No change. Clicked keys on keyboard, no change. Held ON button of Mini to cause a shutdown and reboot. After about 3 tries, finally got a signal to monitor and all operated well until I turned it off. Purchased a new and expensive NEC Multisync EA244wmi monitor, no fix. Tried changing to a HDMI cable, no fix. Tried various energy saver settings, no fix. I have to deal with this 5 days out of 7. What I get when turning it on in the morning is hearing a ding dong, the HD running, but nothing more happens, no Apple logo on the monitor, which shows a no signal alert. When trying to awaken from a sleep, it is a dead brick. In both cases, I keep drying a shutdown and waiting 10 seconds to boot up again. Sometimes it does, sometimes not.
The Apple site support references, about not awakening from sleep are mostly about laptops, are nearly useless. And no helping points on the problem of unable to send signal to monitor when doing an original boot. I have read strings of posts about these issues and tried various ideas and there has been zero improvement. I am very, very disappointed in the MacMini, as I had a MacBook Pro prior which had only 1 problem in 4 years. With my MacMini, right out of the box, that won't send a signal reliably to the monitor, there is nothing to see to do a diagnostic - it is a brick. Rebooting 3 or 4 times, unplugging cables and replugging, when I took it out of the box, in frustration I came close to taking it back to Apple under warranty, a 50 mile trip which is why I have not yet done it. After hassling with this for a few months, I am getting ready to return it under warranty, and to refuse a like replacement, based on what I have read here - and then consider reverting to a MacBook Pro or an iMac.