Some activities need little mental effort such as playing certain video games. With practice, you improve at them. The same idea applies to useful skills like typing or solving maths problems, which have practical benefits. What other activities could someone do that are either helpful for an engineer or generally useful and need little thought?
This isn't quite what you're looking for, but you could label data. It's low effort (usually) and creates something useful, although it doesn't necessarily benefit you personally.
Copy paragraphs from textbooks. Time-consuming but forces one to slow down and think.<p>Memorization mnemonics. Shuffling through index cards to associate numbers to words. See <i>Mind Performance Hacks</i> (2006) from O'Reilly.<p>Make notebooks with cheap filler paper, high-capacity stapler, and duct tape. Use G2 pens or archival quality ink.<p>Make your own index cards: fold in half and then into thirds; cut those into rectangles. (Still on the lookout for good storage.)<p>Shade paper with crayon. Carpet is smoother than desks. Envelopes are a good way to iterate color combinations. Pick colors at random.<p>Convert mailed coupons into CSS. You get salable graphic designs and color palette for free.<p>Break down milk cartons for free cardstock.<p>Prime cardboard with paint for a cheap canvas.
Pick up garbage at a local semi-wild area/walking path. Pay attention to nature and how the seasons change, as you pick up garbage. Enter trees and animals you spot into the iNaturalist app.
Cooking. You start by following recipes, and slowly absorb the techniques and taste into your daily life.<p>You can start cooking relatively cheaply. I do remember having sharp knifes and a good cutting board, some decent pans and you'll have a good time.
Currently it's photography for me.<p>I'm documenting graffiti a lot, but also taking pictures of other urban/nature/landscape scenes.<p>Took ten thousands of photos in the past years with a full-frame camera to the point that wielding the camera is second nature.<p>Also learnt editing raw files with a professional tool and trying to improve on it.<p>I hate taking photos with my smartphone now.
Processing plant fibers for string/rope/etc. Nettle, yucca (so I've heard), hemp, ... it's a long list but regional.<p>Humming to find the resonant frequencies of the space you're in.<p>Observational drawing- paper, pencil, draw what you see (or if not sighted, maybe there's a similar activity?)<p>Listening to what's happening around you. Originally thinking birdsong, but the lowest barrier is just where you are.
Without sweating the details, you could focus a bit on avoiding the annoying high-effort but useless activities that seem to come up too often under emergency conditions if you don't watch out :\