My favorite ENT mnemonic: "7 up, coke down." Refers to the orientation of the nerves in the distal IAC. (given to NPB this is arguably a gross anatomy or ENT sub I level of knowledge)
There are no cancer staging mnemonics that I know of. The good news is that most of the staging systems are similar, and once you understand the anatomy and how the disease behaves, it makes sense why they are how they are. I had melanoma on my oral boards, which is probably the hardest for me to remember, partly because I rarely see it.
Patterns... There are a lot of 2 and 4 cm cut offs for the T stage, hopefully you noticed that for most of the upper aerodigestive SCC. Learn the subsites of the larynx to learn supraglottic staging.
N stage is the same for a lot of primaries. Thyroid is a good exception to make sure you know the alternate N staging.
Then it is just a matter of recognizing the different primaries that have their own unique staging systems . Ehtesioneuroblastoma (Kadish staging system), ear canal & temporal bone (Arriaga modified Pittsburgh staging system), glomus tumors (multiple staging systems), nasopahryngeal ca, sinonasal ca. Even if you don't memorize them all, if you know they exist, you can quickly reference them.