I've just been told by a French speaker you can say "je vais lui montrer" to mean "I will show her". Help! I'm confused!

I suspect the confusion is that when it's a stress pronoun, "lui" always means "him", but as an indirect object pronoun, it's also used for "her".

Using lui/leur = him or her/them (French Indirect Object Pronouns)

Indeed, that's a perfectly fine sentence. Let's take a look at it:

Montrer qqc à qqn. -- To show something to someone.

The object you're showing is the direct object and the person you're showing it to is the indirect object.

Without the direct object, the meaning -- similar to English -- can also be to convince someone as in "I'll show her!"