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Looking back to Opportunity(Rest In Peace, little friend) it was apparently designed to operate for 90 days but it ended up going for 16 years which is approximately 64 times longer than the engineers hoped for. This blows my mind. The technology that we buy and use here on Earth seems so fragile and badly engineered compared to Opportunity.

Besides, Opportunity is not the only one. The second Mars rover, Spirit, was also meant to last for far shorter than it actually did(even though it wasn't nearly as tough as Opportunity). And if I remember correctly, the both Voyagers were also estimated to lose connection with Earth far sooner.

Now of course I admire and appreciate the mental and physical work the engineers had to go through to design a rover that lasts for almost two decades on another planet, but I still don't understand how they did that.

How come so many space probes are able to survive for such long periods of time and why is the difference between the expected duration of service and the actual duration of service so dramatically large? Is it really that hard to predict how long a device will last?

They needed to guarantee that it would operate for its expected duration. Each component has a Mean Time Between Failure (MTBF). The important thing here is that the MTBF is an average. That means that half of the similar components will fail before that time. The MTBF like most statistical measurments follows a bell curve (see below):

In this chart, figure that the horizontal axis measures time and that the MTBF is at the zero position.

The trick is to make the part rugged enough and/or have enough backups (NASA generally goes with the "and" here) that the expected life falls in the -3 range.

That makes it extremely unlikely for the part to fail before its expected life to run out and much more likely to last until a bit after the MTBF.

Every component and group of components is designed this way. That means that anything they build will, in most cases, last much longer than its MTBF.

Very good question! The answer boils down to statistics of failure. Some aspects involve the statistics of "random" failures—for some reason some critical component just bites the dust—and some involve event-driven failures, such as failures induced by landing shocks, long engine burns, atmospheric entry stresses, etc.

When someone (a government, usually) spends hundreds of millions to billions of dollars/euros (or the equivalent in yen, or rupees, or rubles, or whatever) for a scientific mission, they want the probablility of failure to be "acceptably low", which usually means very low. The more is spent, usually the smaller the accepted probability of failure. Typical numbers I have seen working with NASA and JPL are 95% probably of success for a relatively inexpensive mission, and 99% or even higher for flagship-class missions. (Probability of success = 1 - Probability of failure) Pushing to those high success probabilities gets really expensive.

Probabilities of random failures are not exactly normally distributed, but let's treat them as such. To get the expected probability of failure over the mission's intended lifetime to very low values, you have to make time to the 50% probability of failure a lot longer than that, sometimes many times that. You're way out on the wings of a normal distribution. At 95% probability of success, you're 2$sigma$ (two "standard deviations") from the mean, that 50% probability of failure. If you're wanting a mission duration of, say, 5 years, with a 95% probability of success (5% probability of failure) and the standard deviation of failure is 4 years, then you have to design for a mean time to failure of $5 + (2 times 4)$ years, or 13 years. So half the time, you expect this spacecraft designed for a 5-year mission to last 13 years.

Event-driven statistics can modify that further. Components for a lander or rover must be designed to survive the atmosheric entry (for a destination with an atmosphere) and landing. There is a statistical probably that those components will fail, but they have to be designed with the robustness to make that probability very low. But designing for survival during landing often means that, once they've successfully landed, the expected lifetime goes up a lot.

That is true of spacecraft other than landers, too. Spacecraft that are quiescent, i.e. not doing propulsive maneuvers, not doing a lot of radical attitude variations, not running scan platforms rapidly all over the sky, tend to last a long time. This is the case with the two Voyager spacecraft: since Saturn for Voyager 1, and Neptune for Voyager 2, they've largely been in "quiet cruise". Also, a small but dedicated operations team has come up with creative ways of conserving electric power. They figure out such tactics as turning off instruments that are no longer useful, turning off heaters in components that aren't needed anymore, etc. When I was working the Neptune encounter I remember the project saying that they expected to have enough power to operate until about 2015. We've gone well beyond that, mostly due to those power conservation strategies. Suzie Dodd, the Project Manager, says now they're thinking maybe 2025.