So this is probably something I should be able to find in the rule book, but for whatever reason I appear to have failed at that. If you have a card that modifies a runner card (in our specific example it was some doohickey that gave an icebreaker bonus strength) and the card it is modifying is trashed. What happens to the modifying card?

We came to a common sense consensus that it goes away, but wanted to be double-sure, so figured I would ask the esteemed folks here.

So there are two different things that can be happening here depending on the card you're talking about. I'm guessing you're talking about "personal touch." In this case if the icebreaker that is hosting personal touch is trashed personal touch is trashed as well. However you may also be talking about dinosaurus. In that case the icebreaker is actually hosted on the dinosaurus and so if the icbreaker is trashed the dinosaurus stays in play. However if the dinosaurus is trashed the icebreaker that was hosted on the dinosaurus is trashed.

Dug up the actual card, and it is indeed The Personal Touch. Thanks for the comprehensive answer.

What's the actual rule that causes this to happen? Is there a general system for hosted cards?

It's in the rulebook under hosting. Whenever a host card is trashed anything that is hosted on it is trashed as well. Another important part is that the trashing of the hosted card cannot be prevented.