My friend and I are having a debate about this.

The War Caster feat (PHB, p. 170) says:

When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.

I believe this is to eliminate AoE spells. Can I use my reaction to cast Bigby's Hand with the War Caster feat?

My friend's argument is that the spell itself summons the hand, which you can then use one of its effects, and these effects don't necessarily target a creature (example, interposing hand just provides half cover).

My argument is that, since as part of the casting you can choose an effect and certain effects only target one creature, that you'd be allowed to use one of those effects only.

Also, if it's already cast, using one of Bigby's Hand's effects is only a bonus action, whereas the War Caster feat says that you can cast a spell with the casting time of one action. Technically, the spell does have the casting time of one action, but it's already been cast, and the attack would only be a bonus action. Would one of the effects of Bigby's Hand be allowed on the War Caster's opportunity attack if it were already cast?

No to the first question and the second question

With respect to the first question, Bigby's hand (PHB p218) contains the following text.

You create a Large hand of shimmering, translucent force...

Compare this with the text of firebolt (PHB p242)

You hurl a mote of fire at a creature or object within range.

Or Toll the Dead (PHB p169)

You point at one creature you can see within range, and the sound of a dolorous bell fills the air around it for a moment. The target must...

Or Disintegrate (PHB p233)

A thin green ray springs from your pointing finger to a target that you can see within range. The target can be a creature, an object, or a creation of magical force...

All of these spells explicitly mention targeting only one target and should be valid spells for use with War Caster. Bigby's Hand creates a magical effect on the battlefield that may be used in a way that targets only a single target, but it does not pertain to War Caster as it is not a spell that

target[s] only that creature.

With respect to the second question, War Caster states the following (PHB p170)

...you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action...

As you are not using an action and/or casting a spell, using an effect from Bigby's hand that has already been cast does not qualify for use with War Caster.

No

The problem here is the spell description for Bigby's Hand:

You create a Large hand of shimmering, translucent force in an unoccupied space that you can see within range.

The War Caster feat's ability to cast a spell instead of an opportunity attack can only be used with a spell that targets only the creature provoking the attack of opportunity. But Bigby's Hand targets an unoccupied space for the hand to appear in. As such, it won't work with War Caster.