If a pattern calls for a 3mm needle, what US size would you use? I checked on line for conversion charts and can't find one that shows an equivalent for 3mm.
I think it is a US size 2.
There is no exact US equivalent. Try US 2 and 3, and use whichever gives you the correct gauge. If you are growing a stash of needles, a metric 3 is a good addition.