Learn French: Je lui pardonne son mensonge.

In French, the verb pardonner (to forgive) is used with the preposition à when it refers to a person (pardonner à quelqu'un). Because it connects with à, the person is an indirect object. Lui is the indirect object pronoun for "him".

Could lui also mean "I forgive her"?

Yes! As an indirect object pronoun, lui means both "to him" and "to her". You can only tell from the context of the conversation whether you are forgiving a man or a woman.

Why is lui placed before the verb?

In French, object pronouns like lui almost always go immediately before the verb they are attached to. While in English we say "I forgive him", the French word order is literally "I him forgive" (Je lui pardonne).

Does son mensonge mean it has to be "his" lie rather than "her" lie?

No. In French, possessive words like son agree with the noun that comes after them, not the person who owns the thing. Because mensonge (lie) is a masculine word, you must use son whether it translates to "his lie" or "her lie".