Asking Chinese mother-in-law to learn pinyin?
If your mother-in-law confuses n and l, it's unlikely that knowing pinyin would be enough for her distinguish those words like in standard Chinese. Just because she knows what an n or l is doesn't mean that she'll know which words are supposed to begin with what consonant. If you want her to distinguish n/l in her speech (not sure from your question whether you are), you're not just teaching her pinyin; you'd have to teach her the pinyin of all the n/l words, and then have her speak it that way.

For instance, my parents merge s/sh, z/zh, c/ch, and n/ng. They know pinyin, and they know they confuse those sounds, but they don't necessarily know how those words are supposed to be spelled.  Plus, I've told my mom that 弄 (nòng) should not be pronounced lòng, but she does it anyway, even though she distinguishes n/l in almost all other circumstances. Old habits die hard.

For an another analogy, let's say you are a speaker of American English and have a British friend who can't understand it when all those pesky Americans who keep pronouncing "caught" like "cot". Would you really know whether "not", "naught", or "nought" is the one with the /ɑː/ or the /ɔː/?



If you want to understand her better, I'd recommend just getting used to the n/l merger or whatever idiosyncratic pronunciation your mother-in-law uses. When she says n, consider if it could've been an l word, and vice versa. Have her write out the characters on a piece of paper instead of in the air. Perhaps even just have her dictate into Google Translate or Bing Translator.

As for how she can understand you better, consider pulling out your phone and using an IME instead of writing out the pinyin. Even if she learns pinyin, it's probably going to be a lot slower for her to read it than hanzi. When I was first learning hangeul, it took me a few months to get comfortable enough to not mix up some letters. Something with haphazardly designed letters like Latin, and something with quirky spelling conventions like pinyin, are probably going to be pretty hard to read for an elderly person. With regional dialects, she might be getting the tones or other phonemes of the word wrong too, so she might have trouble decoding what you wrote.

Also, maybe ask your wife why she thinks her mother has trouble understanding you. Are you subtly mispronouncing the words somehow (maybe in terms of tone), or does your mother only understand a certain nonstandard pronunciation?