I have an outlet with an open neutral, as determined by a plug in tester, so the outlet does not work. If I ran a separate neutral wire to another a separate group of neutral wires connected with a wire nut, will that work?

An Open Neutral or a loose Neutral can cause a fire because as current passes through the wire it arcs to an adjacent wire.

The ONLY process to use is to identify the receptacle with the loose neutral and correct the connection.

Usually, the last working receptacle will have the connection loose. Pull the receptacle and look for "Back Stabbed" wires or loose wire connection s on the receptacle. Then look at all of the wire nuts (if any) to make sure that all of the wires under the wire nut are tight and that the nut itself is tight.

What don said AND do not try to run another neutral to that outlet, fix the one you have.