Like Mwiess, I would give an example from social sciences or sports or the like of a function with several inputs. I think this is the critical first idea to get across. After this and only after this, you can get to discussing things like interactions of the factors.
I would NOT start with a saddle point as in one answer. This requires thinking in 3D Cartesian space and puts people's back up who are less mathy. Not saying it is not useful example and would use it in a class (it has value, which is why texts use it all the time). But realize than that even with weak students, you are dealing with people who are in a process of learning and used to these structures. With a layman, would not refer to such a structure at first. If you absolutely have to use a visual, just try something like a topo map or talk about mountains. (But even here, I worry that the reference to a Cartesian system (directions and altitude) will bother some people. But worth a shot. Definitely do that before the saddle point though!
Once, you have gotten into useful discussion of the concept of multivariable functions, you have give the person the most useful insights for daily life. Even though you haven't gotten into the diffyq part of it at all. If the person responds well and you want to carry on further, you could mention that PDEs are a method of going from evidence about how the factors change to try to unravel the equation. If the person knows integral calculus, refer to that. Solving an ODE or PDE is kissing cousins to "going backwards" in integration. Asuming they don't, don't refer to it.
The other thing that is extremely useful for a layman is to give them some context of where PDEs occur in a curriculum and who needs to learn them. (Something that I find lacking at times on Stack.) Mention how it is really part of the "calculus track". That it is like "harder calculus". and comes after 3 semesters of calculus and one of ODEs (just say simple DEs if you want). That it is used by engineers and physicists (and of course mathematicians). But I would emphasize the place in the technical curriculum over the math track...after all look at the numbers of students.
P.s. I will even admit to not knowing PDEs well. Have encountered them at times but never really mastered them. But knowing where they "fit in the universe" is still useful. So I have some sympathy for the layman and some feel for what is useful for him to understand.
This is why I struggle with Wikipedia articles about math concepts or math subjects which take a definitional approach (with lots of rigor) as the very first step to understanding. Rather than saying "a lion is sort of like a cat but much bigger. It is yellow and has big teeth. The males have a bunch of hair on their head called a mane. They live in Africa and can kill big animals like deer or zebra and can kill a man no problem." This helps me out way more than some formal taxonomy (with massive amounts of recursions or lookups...in math with an amount of lookups that basically requires learning courses to understand even an overview encyclopedia article!)
An easy example would be Lie groups. I have no idea what they are but would like to know. I don't have a deep math background but way more than the average cat (1580 on old style SATs, math through ODE and some engine math, lots of engineering and science and even some baby group theory in the context of crystallography and IR spectroscopy.) So while perhaps not ready to learn Lie algebra, I am WAY more knowledgeable than a complete layman (say a smart NYT reader who "hated math" and was an English major in college). Yet I can't even just get a basic orientation. So in your discussion with the layperson on PDEs, I would try to give him some orientational description o the field. NOT the first lecture in a course and NOT an attempt to teach most of it an hour (to a person who even lacks prereqs). He just wants to know what kind of animal the PDE is!