For a device that is supposed to display "calibrated" values that stay constant over time (ie. measuring voltages, currents, resistances to within some 0.3% of actual), I don't see value in fiddling with the circuitry/programming inside it after purchase.
As a toy/learning device (like Arduino), sure, open source it and have fun. Is that the purpose of this project? Is it aimed at becoming the first open source DMM just for the novelty of it? Add bluetooth for remote control/monitoring/datalogging, etc. I don't know what else you'd want a DMM to do that existing ones don't already do. I have no imagination nor thought about it much.
I suppose price is a factor, so if the open source meter has features (or can add features over time via community updates) X, Y and Z which only DMMs that cost >$500 have, yet the open source meter only costs 1/2 that (note: I have no idea the price point of this meter), then sure it would be neat and give a real smack to the industry as a whole. "Look at this cool relatively low cost open source DMM that has more features than (insert well-known DMM manufacturer name here) and more features are being added every month for free!" Yeah, that would be cool. I can see value in that for sure.
But once modifications have been made to the meter the trust in output accuracy/lack of drift/whatever decreases, except for the hardcore modder/experimenter who compares with reference equipment to ensure reliability/accuracy after changes have been made. As a basic hobbyist, I doubt I'd want or need to modify a handheld meter nor take advantage of firmware updates nor develop mods of my own. When I pick up a DMM, I want and expect it to "just work and be somewhat accurate" in its readings. If I just fiddled with the circuitry or code (done by me or someone else), I'd still have that feeling of "is this right?" in the back of my mind as I take measurements. I don't want to have to think about that when working on a circuit that is being measured because my focus is on that circuit, not the meter being used.
The only update I'd see as useful would be some sort of accuracy increase due to better code algorithms that had been thoroughly tested across many samples and statistics to back up the claims. Only then would I consider upgrading my "testing device that is supposed to be reliable and accurate". But due to the limitations of the chosen chipset/hardware, I doubt such a thing would happen in reality because the percentage gains would be rather tiny.
If the open-source stuff was limited to everything outside the measuring circuitry and code, then obviously I'm worrying about the wrong thing here and you can just ignore what I said. Adding fog lights, ground effects and a rear spoiler to a DMM could be great fun knowing the engine is still going to pass emissions.
All that said, I may not be the intended audience for this type of product. I totally get that. I see a DMM as more of a tool than a plaything. It has one job to do, kind of like a microwave. Sure, an open source microwave could act as a calculator (if it has the full keypad like most do), but I'd not have any interest in that. If I had an idea of a use for open-sourcing a DMM, like adding bluetooth to it for remote control because of "reasons", then sure, this project would be of very high interest to me. But I don't.