Creating genuinely new mathematics is, of course, non-trivial, but you can surely get away with doing less than that while also doing something more creative than just shuffling notation. Maybe your people have some ideas that are important to them but seem weird to us now, like wanting to construct everything by ruler and compass (in fairness to the Greeks, this seems pretty reasonable as a rigorous basis for synthetic geometry--but now that we have the power of algebra and analytic geometry, it seems quaint). The Greek emphasis on geometry also led to centuries of people feeling weird about writing things like "x + y^2": after all, if y^2 represents an area and x a line segment, it doesn't make much geometric sense to add them. And so on. Being stuck with theorems we already know doesn't seem like a terrible problem to me, since what's really interesting is how they are arrived at, conceptualized, and assembled into a coherent whole. Thinking that you can't do anything interesting because you'll still end up with pi = 3.1415926535... seems like worrying that you can't create an interesting language because you'll still end up being able to say "the dog bites the man".
You might also try to imagine things like: what if the Greeks had thought of probability? What would the next 1000 years of mathematics look like? Basic probability / combinatorics is simple enough that I think this is entirely reasonable, and really it's kind of a surprise to me that it took so long for such ideas to come about! Didn't people ever roll dice? Of course this also leads you back to the idea of expressing old facts in new ways. Eudoxus and Archimedes and so on anticipated the integral calculus with the geometric "method of exhaustion" for computing areas, but it's easy to imagine also approaching these ideas via, say, probability. The law of large numbers is pointing towards limits, for example, or you might think of measurements in a probabilistic way: the area of a shape inside a 1x1 square is the same as the probability that a dart thrown at the square hits the shape.
Also, do not forget that the things you know now were almost certainly not arrived at in their final, nicely polished form. Give your conpeople glimpses of some bigger ideas, different glimpses than we had historically. There are lots of ways to arrive at the same idea, not all of them obvious. Nowadays (at least in America), kids tend to first be exposed to complex numbers in the context of solving quadratic equations: sometimes the quadratic formula tells you to take the square root of a negative number. But no one cared about that 500 years ago: a solution of a quadratic equation likely represented some geometric thing, and there was no reason to care about ???square roots of negative numbers??? (indeed, you probably threw out negative solutions too). The first place it became apparent that complex numbers really mattered was after Tartaglia/Cardano/etc. worked out how to solve
cubics. For example, the cubic x^3 - 7x^2 + 14x - 8 = (x-1)(x-2)(x-4) has roots 1, 2, 4. But the cubic formula gave Cardano, for one of the roots, something like:
This turns out to be equal to 1. Cardano may not have been interested in non-real solutions to his equations, but he certainly would be interested in getting "1, 2, 4" here, and so for the first time someone was forced to care about complex numbers.