Suppose an algebraic variety over Q, or subvariety of one, has a parametrization, also over Q.
Clearly an infinite number of birationally equivalent parametrisations can be obtained from this. But some will be better than others in respect of small-height rational points being consistently obtainable from small-height values of all the parameters.
I wondered if there is a name for this property, such as (off the top of my head) "well conditioned", or "balanced", or something along those lines, and maybe even a quantitative theory (an aspect of Diophantine Approximation perhaps) of how well a parametric solution can have this "balanced" property for all the rational points it includes.
For some higher-dimensional [sub]varieties, I imagine it may be that no single parametric solution is optimal in this sense for all rational points, any more than (using a metric metaphor) no single point is "near" all the others.
edit: Following added for clarification:
As an example, the surface defined by $x^2 + y^2 - z^2 = 1$ has a simple rational parametrization $x = (u v - 1)/(u - v)$, $y = (u + v)/(u - v)$, $z = (u v + 1)/(u - v)$, where $u, v$ are two parameters (which can each be expressed in terms of "x, y, z").
Obviously, these parameters can be replaced by any other pair related to them by an invertable affine transform with rational coefficients (or more generally by birational transforms, but an affine transform is sufficient as an example).
For example, letting $u = 1000 p + q$, $v = 999999 p + 1000 q$ gives an equally valid general parametrization in terms of $p, q$; but clearly this is less suited to obtaining small-height rational points (x, y, z) for small-height rational parameter values, because the "lopsided" transform throws the parametrization out of whack in that respect.
I agree that the concept, as far as have been able to sketch it, may seem somewhat wooly, and possibly not very useful. But I believe a notion of entropy can be defined for sets of birational transforms. So perhaps that and related ideas may have some bearing on the question.