What star is the closest to Betelgeuse? Based on this answer, I should be able to figure out this question, if I know the distance between Betelgeuse and its nearest neighbor. So what is Betelgeuse's nearest neighbor? 

I do mean the nearest neighbor in 3D space, not in the plane of the sky. Since Betelgeuse is fairly close to the Sun, I'm guessing we should know its neighborhood with a pretty good precision.
$$
\begin{array}{|c|c|c|}
\hline
\textbf{Star} &amp; \textbf{Magnitude} &amp; \textbf{Distance (ly)} \\
\hline
\text{Betelgeuse} &amp; \text{0.45} &amp; \text{0.00} \\
\hline
\text{HIP27648} &amp; \text{8.24} &amp; \text{17.82} \\
\hline
\text{HIP28478} &amp; \text{8.42} &amp; \text{19.03} \\
\hline
\text{HIP27573} &amp; \text{7.41} &amp; \text{24.27} \\
\hline
\text{HIP28175} &amp; \text{9.1} &amp; \text{39.34} \\
\hline
\text{HIP26887} &amp; \text{7.28} &amp; \text{39.34} \\
\hline
\text{HIP29261} &amp; \text{7.66} &amp; \text{41.42} \\
\hline
\text{HIP26914} &amp; \text{9.6} &amp; \text{41.87} \\
\hline
\text{HIP27513} &amp; \text{8.28} &amp; \text{42.65} \\
\hline
\text{HIP27386} &amp; \text{5.26} &amp; \text{44.20} \\
\hline
\text{HIP28261} &amp; \text{8.45} &amp; \text{46.08} \\
\hline
\text{HIP28686} &amp; \text{6.09} &amp; \text{46.73} \\
\hline
\text{HIP27004} &amp; \text{7.89} &amp; \text{47.16} \\
\hline
\text{HIP29537} &amp; \text{7.82} &amp; \text{49.28} \\
\hline
\text{HIP27324} &amp; \text{9.22} &amp; \text{49.88} \\
\hline
\text{HIP27932} &amp; \text{8.26} &amp; \text{50.41} \\
\hline
\text{HIP29868} &amp; \text{8.92} &amp; \text{50.69} \\
\hline
\text{HIP28397} &amp; \text{8.24} &amp; \text{50.95} \\
\hline
\text{HIP29236} &amp; \text{6.57} &amp; \text{52.93} \\
\hline
\text{HIP27902} &amp; \text{6.3} &amp; \text{52.98} \\
\hline
\text{HIP28232} &amp; \text{6.22} &amp; \text{54.81} \\
\hline
\text{HIP27315} &amp; \text{7.14} &amp; \text{55.28} \\
\hline
\text{HIP27751} &amp; \text{6.9} &amp; \text{55.37} \\
\hline
\text{HIP30124} &amp; \text{9.75} &amp; \text{56.46} \\
\hline
\text{HIP29685} &amp; \text{7.69} &amp; \text{56.79} \\
\hline
\text{HIP27412} &amp; \text{9.31} &amp; \text{58.80} \\
\hline
\text{HIP27153} &amp; \text{7.46} &amp; \text{60.13} \\
\hline
\text{HIP28202} &amp; \text{7.97} &amp; \text{60.42} \\
\hline
\text{HIP27919} &amp; \text{7.15} &amp; \text{60.85} \\
\hline
\text{HIP28056} &amp; \text{7.17} &amp; \text{60.89} \\
\hline
\text{HIP29381} &amp; \text{8.88} &amp; \text{61.26} \\
\hline
\text{HIP26986} &amp; \text{9.17} &amp; \text{61.26} \\
\hline
\text{HIP29630} &amp; \text{8.15} &amp; \text{62.56} \\
\hline
\text{HIP29590} &amp; \text{7.82} &amp; \text{64.37} \\
\hline
\text{HIP30120} &amp; \text{6.76} &amp; \text{64.57} \\
\hline
\text{HIP26795} &amp; \text{6.82} &amp; \text{66.03} \\
\hline
\text{HIP26655} &amp; \text{7.09} &amp; \text{66.99} \\
\hline
\text{HIP26615} &amp; \text{9.13} &amp; \text{67.50} \\
\hline
\text{HIP27895} &amp; \text{7.04} &amp; \text{68.80} \\
\hline
\text{HIP27309} &amp; \text{9.7} &amp; \text{69.04} \\
\hline
\text{HIP28323} &amp; \text{8.16} &amp; \text{69.12} \\
\hline
\text{HIP28171} &amp; \text{8.68} &amp; \text{69.91} \\
\hline
\text{HIP25767} &amp; \text{8.93} &amp; \text{70.26} \\
\hline
\text{HIP25698} &amp; \text{8.74} &amp; \text{70.31} \\
\hline
\text{HIP26107} &amp; \text{7.83} &amp; \text{71.05} \\
\hline
\text{NuOri} &amp; \text{4.42} &amp; \text{72.70} \\
\hline
\text{HIP27284} &amp; \text{8.72} &amp; \text{73.29} \\
\hline
\text{HIP27968} &amp; \text{8.42} &amp; \text{73.36} \\
\hline
\text{HIP29599} &amp; \text{7.96} &amp; \text{73.41} \\
\hline
\text{HIP25424} &amp; \text{8.63} &amp; \text{73.44} \\
\hline
\end{array}
$$

Subject to important caveats below, the table above lists the 50 stars closest to Betelguese including Betelguese itself:


I ignore my own advice and used HYG data to create this list.
The repl.it code: https://repl.it/@barrycarter/Twitch-BeetleJuice
I hope to turn this into a webapp later; for now, by changing the var fixedStar = stars[27919]; line in script.js, you can run this for any star in the HYG catalog (be sure to use the HYG id). You can convert to LaTeX using https://barrycarter.github.io/pages/CSV2LATEX/
As others have noted, the distance to Betelguese is not accurately know: "https://en.wikipedia.org/wiki/Betelgeuse#Distance_measurements". Of course, the same may apply to other stars in the HYG catalog, including those I have listed above. Therefore, the computed distance between Betelguese and another star may be way off.


I'm hoping to get a "better" list by using GAIA DR2 data, and will amend this answer when I do. Of course, while GAIA DR2 lists more stars, the distance measurements can be even further off.