Hello!
Let me just chime in a bit. First, if you don't have it, buy (or make) a shortcut plug like the one in photo.
You can use it to check zero volt offset, zero ohm offset and also small current range offset.
What you see here is probably induced interference, it's nano amperes... Everything should be shielded, your 1V source should be battery powered, shielded cables and such.
And than, last two digits on ANY 6.5 voltmeters are a finicky bunch.
Look here on Metrology section there a several topics of measuring and sourcing nano and even femtoamps...
All the best,
Sinisa