From the algorithm's author (see FAQ section):

How should I set the perplexity in t-SNE?
The performance of t-SNE is fairly robust under different settings of the perplexity. The most appropriate value depends on the density of your data. Loosely speaking, one could say that a larger / denser dataset requires a larger perplexity. Typical values for the perplexity range between 5 and 50.

What is perplexity anyway?
Perplexity is a measure for information that is defined as 2 to the power of the Shannon entropy. The perplexity of a fair die with k sides is equal to k. In t-SNE, the perplexity may be viewed as a knob that sets the number of effective nearest neighbors. It is comparable with the number of nearest neighbors k that is employed in many manifold learners.

There's also this paper on automatic selection ot t-SNE perplexity.

Though not super in-depth, this post is a good primer: How to Use t-SNE Effectively

In most cases, the default value of the perplexity parameter in t-SNE is 30. The standard range for the perplexity parameter is between 10-100. But, you can change this value based on the size of your datasets. In the context of scRNA-seq, setting a perplexity value of 1% of sample size (number of cells) could be useful to preserve the global geometry and in this article, you can see how you can use this perplexity parameter for a very large dataset (say millions of cells). Besides, other parameters such as the number of iterations, learning rate, and early exaggeration factor can also affect the visualization and should be optimized for larger datasets. I have covered more detail here with example https://reneshbedre.github.io/blog/tsne.html

In practice, I think you should just try a few values and see which one(s) "makes sense". It's a hand-waving suggestion of course but after all, t-SNE is a visualization method so there isn't really a correct or optimal way of doing it, there is no statistical hypothesis or test involved. It depends on what you want to show... (the paper linked by Jean-Karim is interesting though and I haven't read it yet).