What is wrong with the following simple definition? Let $g_1$ and $g_2$ be two independent standard Gaussian variables. For $t=(a,b)\in S^1$ (so $a^2+b^2=1$), let $B_t=ag_1+bg_2$. You get a Gaussian process whose distribution is invariant under rotation. Each $B_t$ is a standard Gaussian variable and the variance of $B_t-B_s$ is $\|t-s\|^2$.
Following Ori's remark below, concerning the 2-dimensionality, maybe a better simple suggestion is the following: Take two independent Brownian motions on $(-\infty, \infty)$, $C_a,D_a$ ($C_0=D_0=0$) and for $t=(a,b)\in S^1$ define $B_t=C_a+D_b$.
On third thought, this is probably just a Brownian motion on $R^2$ (zero at the origin) restricted to $S^1$.