Shakuhachi Tips
I believe you cannot know a honkyoku piece deeply until you go through the process of memorizing it. To memorize a piece is to become intimate with it, to explore its body and its bones. Once you memorize a piece, you are free to play with it rather than always trying to capture it.
There are many tricks to memorizing a piece. First, find the sections. Though they may not be marked, most pieces contain a shirabe (introductory) section, a honte (original, main) section, a takane (high, climactic) section, and a final musubi (conclusion). There may also be a hachigaeshi (returning the bowl) section between the takane and musubi. Knowing the particular character of each of these sections can help you memorize them.
Some pieces, for instance the Jin Nyodo version of Koku, are comprised of sections that are more similar to a dan-mono structure. In other words, there are several sections of roughly the same length that are each a variation of the same basic material.
Next, discover the vocabulary of the piece. Does it have a section centered around a koro-koro? Does it use some of melodic fragments or specialized techniques you are already familiar with? What sorts of things does it leave out?
Most importantly, find the patterns. There are small patterns: repeated phrases, fragments that are bookended by set melodies, phrases that repeat but with slight variation. There are larger patterns as well: entire sections that are bookended, or sections that act as bookends to the main body of the piece. Some pieces, like Kyorei, remain emotionally static but follow a very methodical pattern of variation throughout. Once you discover a pattern like this, memorization becomes very easy.
In addition to patterns, there may be a story that can help with memorization. Tsuru no Sugomori, for instance, is very difficult to memorize unless you follow the narrative of the nesting cranes.
Here are some pieces to get you started. Though their technical level varies, they are among the easiest to memorize due to simple patterns or adherence to the sectional divisions outlined above:
Choshi
Kyorei
Daiwa Gaku
Kumoijishi
Banshiki
Sanya Sugagaki
Hi-Fu-Mi Hachigaeshi
Koku
Banshiki-cho
Reibo
Shika no Tone