An exercise (8-12) in Lee's Introduction to Smooth Manifolds involves showing that if $F : \mathbb{R}^2 \to \mathbb{RP}^2$ is given by $F(x,y) = [x,y,1]$, then there is a vector field on $\mathbb{RP}^2$ that is $F$-related to the vector field $X = x\partial/\partial y - y\partial/\partial x$ on $\mathbb{R}^2$.
I solved this problem as follows: We begin by letting $U_1,U_2,U_3 \subset \mathbb{RP}^2$ be the open subsets on which the first, second, and third coordinates, respectively, are nonzero, and let $(u_i,v_i) : U_i \to \mathbb{R}^2$ be the usual coordinate systems for each $i = 1,2,3$. We then define a smooth vector field $Y_i$ in coordinates on each $U_i$ as follows: \begin{align*} Y_1 &= (u_1^2 + 1)\frac{\partial}{\partial u_1} + u_1v_1\frac{\partial}{\partial v_1} \\ Y_2 &= -(u_2^2 + 1)\frac{\partial}{\partial u_2} - u_2v_2\frac{\partial}{\partial v_2} \\ Y_3 &= -v_3\frac{\partial}{\partial u_3} + u_3\frac{\partial}{\partial v_3}. \end{align*} It's then a straightforward computation with Jacobians to show that these three vector fields agree on intersections, and so they extend to a smooth global vector field $Y$ on $\mathbb{RP}^2$. One more computation shows that $Y$ is $F$-related to $X$. (I might have made a computational error here but that's beside the point.)
Despite having a formula for the vector field $Y$, I still have no intuitive grasp of what it actually looks like. $\mathbb{RP}^2$ is already a pretty abstract object, and how to imagine vector fields on it is a mystery to me---the above coordinate representations don't shed that much light on its structure. Is there a coordinate-independent way to define $Y$? I'm thinking maybe we can define a visualizable vector field on $\mathbb{R}^3 \setminus \{0\}$ that sinks through the quotient map $q : \mathbb{R}^3 \setminus \{0\} \to \mathbb{RP}^2$, but I don't know how the details would work out.
So you have the right idea in trying to define a vector field on $\mathbb{R}^3 \setminus \{0\}$. The main tool that you need is problem 8.18(c), which says that if $F \colon M \to N$ is a surjective smooth submersion for positive-dimensional smooth manifolds and $X$ is a vector field on $M$, then there is a unique vector field $Y$ on $N$ that is $F$-related to $X$ iff $dF_p(X_p) = dF_q(X_q)$ whenever $F(p) = F(q)$. (See this answer for details.)
Then it's straightforward to verify that $\pi \colon \mathbb{R}^n \setminus \{0\} \to \mathbb{R} \mathbb{P}^{n-1}$ sending $(x^1, \dotsc, x^n)$ to $[x^1; \dotsc; x^n]$ is a surjective smooth homomorphism. (See this answer for the $\mathbb{C}$ case, which is very similar.)
Then it's a straightforward computation in coordinates to show that if a vector field $X$ on $\mathbb{R}^n \setminus \{0\}$ is homogeneous, i.e. $X_{\lambda p} = \lambda X_p$ for all $p \in \mathbb{R}^n \setminus \{0\}$ and $\lambda \ne 0$, then the conditions of 8.18(c) are satisfied and there is a unique vector field $Y$ on $\mathbb{R} \mathbb{P}^{n-1}$ that is $\pi$-related to $X$.
Also, you'll need Prop. 8.23, which says that if $S$ is immersed in $M$ and $Y$ is a vector field on $M$ tangent to $S$, then we can restrict $Y$ to $S$ such that $Y$ is $\iota$-related to $Y|_S$, where $\iota$ is the inclusion map of $S$ in $M$.
Finally, it's straightforward to show that if you have $M \xrightarrow{F} N \xrightarrow{G} O$ and vector fields $X$, $Y$, $Z$, on $M$, $N$, and $O$ respectively such that $Y$ is $F$-related to $X$ and $Z$ is $G$-related to $Y$, then $Z$ is $(G \circ F)$-related to $X$.
Then let $M = \mathbb{R}^2$, $N = \mathbb{R}^3 \setminus \{0\}$, and $O = \mathbb{R} \mathbb{P}^2$, and let $Y = x \, \partial_y - y \, \partial_x$ be a vector field on $N$. Let $F$ send $(x, y) \in M$ to $(x, y, 1) \in N$, and let $G = \pi$ from above. Then it's straightforward to verify that $M$ is immersed in $N$ (embedded, even) via $F$ and $Y$ is tangent to $M$, so $Y$ restricts to $X = x \, \partial_y - y \, \partial_x$ on $M$. It's also straightforward to verify that $Y$ is homogeneous, so there is a unique $Z$ on $O$ that is $G$-related to $Y$. Therefore, $Z$ is $(G \circ F)$-related to $X$. But $(G \circ F)$ is precisely the map given in 8.12, so $Z$ is the desired vector field on $O = \mathbb{R} \mathbb{P}^2$.