Where are your pictures?

I'm not sure how prolonging the life of a subatomic particle constitutes time travel.

He didn't say that it would disappear and show up again 10 minutes later, he said that prolonging the life would illustrate that it traveled through time. That's more of an indirect observation supported by hypothesis, not exactly Occam's Razor.

And let's just say for a nanosecond that what this guy is saying is true, and this neutron did "travel through time", well, what's the big deal?

First of all, it's a simple neutron. It's not a human being. A neutron cannot place a bet, and it can't warn your dad about smoking. Neutrons are among the simplest of the universe's building blocks. Human beings are slightly more complicated to take apart and put back together. How many people will this guy's machine depixellate before they start to get it right?

Second, he's supposedly "sending" it forewards. If you're really time traveling here, send the thing backwards in time and have a neutron sitting there before you start the experiment.