Consider a balloon with no tie. Like a basketball or something. The point is that the surface is elastic and the sphere is expanding. Balloons make the best metaphor for that imo. If you can think of a better one I'd be glad to hear it.
Only other thing I can think of is a "rubber bubble" which... is a balloon.
I often think of it as a soap bubble, but I have to stop and check myself: are we choosing spherical because that's simple, or because we have evidence? It could be toroidal or something even more bizarre.
I see it as a sphere because of the uniform expansion of the Universe. The Universe might just be so large that any measurable difference in the rate of expansion is outside the observable Universe, which could suggest a shape other than a hypersphere, but I don't subscribe to that school of thought.
Only other thing I can think of is a "rubber bubble" which... is a balloon.