First off, I'm not sure that the assertion that ${\rm H}^1(-,\mathbb{Q}(n))$ doesn't change under purely transcendental extensions is known. The Gersten complex for the affine line provides the Milnor exact sequence
$$
0\to {\rm H}^1(F,\mathbb{Q}(n))\to {\rm H}^1(F(T),\mathbb{Q}(n))\to \bigoplus_{x\in (\mathbb{A}^1)^{(1)}} {\rm H}^0(\kappa(x),\mathbb{Q}(n-1))\to 0
$$
The vanishing of the groups ${\rm H}^0(\kappa(x),\mathbb{Q}(n-1))$ is only known for $n=2$ because we understand $\mathbb{Q}(1)$ in terms of units. For $n>2$, the birational invariance would assume the Beilinson-Soule vanishing conjecture which isn't generally known (but ok for finite and global fields).
A significant part of the motivation seems to come from the rigidity of regulators. This is discussed for example in Section 7 of
- D. Ramakrishnan. Regulators, algebraic cycles, and values of L-functions. In: Algebraic K-theory and algebraic number theory, Contemp. Math. 83, Amer. Math. Soc. 1989, pp. 183-310.
The conjecture is formulated in 7.1.8 (but the remark concerning indecomposable $K_3$ seems incorrect). The main point is that for any algebraically closed field $F$ with $\overline{\mathbb{Q}}\subseteq F\subseteq \mathbb{C}$ the image of the regulator map on ${\rm H}^1(F,\mathbb{Q}(n))$ in Deligne cohomology coincides with the image of ${\rm H}^1(\overline{\mathbb{Q}},\mathbb{Q}(n))$ (which we know from Borel's computations). This follows from the argument for Assertion 2.3.4 and a rigidity statement in Lemma 1.6.6.2 in
- A.A. Beilinson. Higher regulators and values of L-functions. (english translation) Journal of Soviet Math. 30 no.2 (1985), 2036-2070.
In the specific case of ${\rm H}^1(F,\mathbb{Q}(2))$ (indecomposable $K_3$) we can express the regulator in terms of the dilogarithm which has a rigidity property formulated in Corollary 6.2.2. of Bloch's Irvine notes:
- S. Bloch. Higher regulators, algebraic K-theory and zeta functions of elliptic curves. CRM Monograph Series 11, Amer. Math. Soc. 2000.
Similar rigidity properties are also true for the polylogarithms, cf. Theorem 50 in chapter 12 (Function theory of polylogarithms) of L. Levin's "Structural properties of polylogarithms". So this ties in with the picture that the regulators should be expressed in terms of polylogarithms, together with the expectation that the regulator should be injective on ${\rm H}^1(F,\mathbb{Q}(m))$.
Formulated differently, the rigidity conjecture (say for characteristic zero fields) would be a consequence of the Beilinson conjecture: any element of ${\rm H}^1(F,\mathbb{Q}(n))$ would always come from some finitely generated arithmetic scheme over $\mathbb{Z}[1/N]$ and then rigidity for motivic cohomology would follow from a similar rigidity property of Deligne cohomology.
