The problem is almost certainly a failure in the “zebra connector”— an elastomeric pad with embedded connectors that’s held by compression between two of the internal circuit boards in the calculator. Over the decades the elasto- part degrades, leading to intermittent or failed contacts that can result in the dead display you’re seeing.
Fortunately the HP-41 series is held together with screws, not sonic welds, so it can be disassembled and often repaired. However, the zebra strip is not the only thing that can degrade with time— the plastic of the case can become brittle and the posts the screws tap into will often fracture and fail with repair attempts.
You have three choices that I can see:
1. The best choice is to simply buy a Swiss Micros DM-41X. It’s a modern interpretation of the HP-41 running as an emulator on an ARM CPU. With a stainless steel case, giant multi-line display, a great keyboard that reproduces the classic HP keyboard feel, and about a zillion times the performance of the original, it’s the go-to choice for engineers and hobbyists that actually use their calculators. The downside is that it’s rather expensive at almost $300.00:
2. Send your existing calculator in for professional repair:
3. Buy a modern replacement for the zebra strip and repair it yourself:
I’ve worked on some old HP-41s in the past and depending on the condition of the plastic— specifically how friable it is— such repairs can range from “Gee, that was easy!” to simply throwing up your hands and accepting that it’s a total loss. There are some techniques that can save even badly degraded cases but they’re beyond my minor technical skills.
In any case, good luck with whatever you choose.
By the way, did you know there’s an annual conference for users of HP calculators?