I want just for fun and because it can be done build own DMM with reasonable resolution and features
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They have nice metering family MSP430i2xx with up to 4 24bit SDAC, they are cheap with reasonable precision, whole chip cost like alone 24 bit ADC or just similar spec voltage reference
I didn't know there is a MSP430 family with an ADC of more than the usual 10 bits, thanks for mentioning it.

The project idea looks like a lot of fun!

Not only fun. It would be a very useful tool for day to day use in low voltage measurements. For sure there is a lot of people that don't need a DMM for electricians (HV capable and with lots of protections and certifications), but rather a fast and versatile low voltage DMM that can be easily connected to a PC (but floating and battery powered).
I'm always annoyed by DMMs. No matter the brand and the price, so far none is perfect. Some are too slow, some don't have a PC interface, some can't disable autorange, some don't remember the previous settings, some have an entire OS and take many seconds to start, some don't have enough internal memory or an SD card for logging, and so on, almost none is perfect for how I would need it to be. Not to mention fancy things like voice reading, display DC and AC ripple simultaneous and with a zoom-able graphic trace, and so on, or the possibility to customize, download or develop new features.
Maybe some interchanging modular hardware would be nice, too, and keep the PCB development effort low. For example stackable boards with distinct functional blocks:
- front end dividers/analog section,
- digital module with MCU,
- display module,
- analog I or R to voltage conversion modules,
- floating PC communication module,
- shield only module/PCB, and so on.