How do you get non-technical folks to appreciate a non-UI problem?

The surest way to get upper level management to buy off on development work is to present it in a quantifiable way. Ideally this quantifiable measure is in $$. You need to explain to them the consequences of skimping on data integrity, security, transactions, etc. and how that will affect the customer\user community and eventually the bottom line. You should be careful in these situations because sometimes management expects these non-functional requirements to "just work." If this is the case, you should either estimate high and work on these items alongside the visible UI work (ignorance is bliss) or you need to document these areas of need as you communicate with management so if things do go bad as you anticipate, it's not your job that is on the line.

Every craft has its unsexy sides. Things that HAVE to be done, but nobody notices them directly. In a grocery store somebody has to organize how and when to fill the grocery shelves so they always look fresh. In a laundry you need somebody who thinks about how the processes should be optimized so that the customer gets his clothes in time.

The tricky part is: The customer won't notice when these subtle things have been done right UNTIL HE NOTICES THEY ARE MISSING! Like when the laundry is not ready on time but two days late, or the veggies in the super market have brown spots and look terrible.

Same goes for IT. You don't notice good transactions until your major customer knocks on your door and tells you that an important and expensive project has failed because the database entries of your product were mysteriously mixed up. You don't notice good security until customer credit card information shows up in Elbonia (and soon after word is in the national newspapers warning customers of your company).

The thing you really have to hammer in again and again and again is that software is NOT static. It has to be cared for even after its initial development phase is over. It is not just a product you buy once and forget about. Every car manufacturer knows that services is of prime importance to the products they build, simply because things WILL occur that have to be fixed and improved. It's the same with software.

So make a presentation, visualize, verbalize, translate your technical information into benefits. Business people don't care about your wish for code aesthetics in a refactoring project, but they WILL understand that your changes will help the product to become more reliable, gain a better reputation and reduce the amount of future service requests. Make them understand by showing them the benefits!

Same thing folks have been doing for thousands of years: draw pictures. Diagram the problems, use visual metaphors familiar to your audience, drag the problem into their territory.

Assuming they're not being intentionally obtuse...

A big +1 for analogies and metaphors. If possible, find one that will resonate with the personal interests of your audience (if it's 1-2 people). For general metaphors, I often find myself using commuter traffic or subways, for some reason.

e.g. We are currently migrating an app from an OODB to Postgres/Hibernate: the bulk of this work is done in Release '4'. Many domain experts have been asking why there are so few user-facing features in R4. I regularly tell them that we have been 'tearing up the city to put in a subway. It is very expensive and undeniably risky, but once it is done, the benefits in R5+ will be astounding, truly.' The true conversation is more involved, but I can return to this theme again and again, well after R4. Months from now, I hope to say "You asked for X and it is now very easy -- precisely because you let us put in that subway back in R4".