Write up a bill for all hours you were there, at the normal rate. Then discount at your discretion. It appears you know him a bit and factor that in if you want.

In summary, keep it simple and clear. Trying to be creative to illustrate a billing method creates the thing you'd be trying to avoid.
KISS

And good on you, Mr Fred!

I would expect to pay a fair amount/wage for your help. NOTHING is free or cheap in this world that is worth having. A professional's time is just that and worth money. I'd either want to pay up front for it or work a trade.
I have mad carpentry/cabinetry skills but I'm no mechanic. I did a similar for my SC a few years ago. I helped remove and install my engine, it went to a rebuilder that I paid XX for and my mechanic and I re-installed and tuned it. I paid both my mechanic and the rebuild guy what was agreed plus what when over budget as well.
AND for no reason this year when I was teaching at a local CCollege how to make countertops I stopped and measured for one as a transaction counter at my mechanics. Did it/dropped it off to pay it just a little forward for having the opportunity to save money a few years ago.

The original post is sort of non-specific.

It sounded like a "good will" situation was originally intended for strangers.
The OP did his part to the best of his abilities, there occurred the expected complications, and the end result was more than the usual headache.

How do you put a value on doing the right thing?
OP should never stop doing the right thing.

1). Bill the customer the total of your actual time as agreed, and prepare for disappointment.

2). Offer customer a work-payment plan (i.e. crap jobs that need to be done around house) for your extra time spent.
It's an opportunity to show them the real nitty-gritty of the business.

3) Or, collect your basic core business money.
And write off the rest as noncollectable, if all else fails.

I'd gladly pay for such an experience.

Oddly enough, I did a complicated framing job for a mechanic friend. Saved him a good 10,000 dollars and never charged him a cent. I'm still cool about it, but he got weird anyway, so don't think that not charging for your services will always make you a hero.

Charge a fair price, and keep things even between the two of you. That's the way of today's world.