How to deal with nagging people?

My mother loves when I buy her gifts. This keeps her in a good mood.

While you are obviously not your mother's husband, the Buddha said in DN 31 a husband should regularly buy their wife a gift (because this keeps their wife feeling loved & appreciated).

The principle applies to sons. When sons give their mothers gifts, the mother feels loved & appreciated.

Well, you can always watch her behaviour (including nagging and other annoyances) as your stored karma. Then to be free from it, as with all kind of karma that I can think of, one must observe it "sharply" (mindfully) along with any secondary phenomena/feelings that arise. Thus watching "sharply", or being mindful, one will not attach self to it, and consequently, remain equanimous in the midst of that nagging. I believe this could have been advised in the suttas at many places, however, I do not have any exact references as of now. Good luck!

You need to grow up and realize what you have in your life. A lot of people in the world never had a mother or a father. A lot of people didn’t grow up with a lot of money or choices. You sound like a spoiled little brat to be honest. This is your mother. Treat her with the utmost and highest respect, love and care in the world.

She won’t be here forever. She might not be here tomorrow, next week or next year. Actually anyone of us can die at any moment. Try to remember that next time you’re with her.

Lastly, remember to be mindful when being with her. Cherish her and buy her a meaningful gift and tell her how much she means to you. Do that as often as you can. She deserves to hear that.

To me it doesn't sound polite or kind to "deal with" family like you "deal with" a nuisance or a chore.

Your description of the problem sounds "conceited" as described here and here.

So I was wondering today what would Buddha do with such people?

I think he left home.

So far as I know the Buddha used to talk with people, or not talk -- and perhaps not "do" very much?

He did give advice to laypeople, there is this whole book which someone lent me once ...

• The Buddha's Teachings on Prosperity: At Home, At Work, in the World
  (by Bhikkhu Basnagoda Rahula)

... which might help, it's all advice to laypeople (a lot of the Buddha's advice was for monks). It includes advice about family relationships, I think it's an accurate summary of suttas sorted by subject. I read it (a few years ago) and found it good. I don't have a copy so I can't quote it but I think you can buy it easily.

Any advice on such people from the Suttas or personal experience?

Personally if ever I feel a moment of annoyance, especially with family, I view that a (temporary, unpleasant) mistake on my part -- unskillful, rude (or childish, immature), hopefully quite rare, and a symptom of heedlessness.

More specifically, if ever I am driving I believe it's of utmost importance that I control myself and the car. I won't drive when intoxicated -- including "intoxicated by anger".

The Dhammapada says ...

222. He who checks rising anger as a charioteer checks a rolling chariot, him I call a true charioteer. Others only hold the reins.

... but I don't entirely trust myself to "check" my anger after I am "feeling" angry. If the sequence is:

1. Irritation
2. Rising anger
3. Anger "checked"

... the I may want some "alone" time or low stress (a walk, a nap, some work, or simply change the subject of conversation) to let the anger (physiological/chemical/hormonal reaction aka "poison") subside.

If I'm trapped in a room with someone who's upset, i.e. angry and loud, experience tells me that can be contagious after a while, i.e. begin to upset me too. Very occasionally in my life I have stopped the car (where it's safe to do so) because my passenger was having a tantrum.

This might have happened more often in my life if I had been driving children! I'm pretty sure my parents sometimes threatened to stop the car, when my brother and I were very young; maybe all parents do.

But driving with adults, my family's convention has been that it's the driver's job to mind the road and avoid accidents -- when I'm driving I sometimes ask for talk to stop when I need all my attention (like when merging onto a highway). A passenger then is free to tell a story if they want to -- or to navigate, read the map, and tell me which turn to take (in advance i.e. with plenty of time to do that safely, not at the last second).

I think that's fair.

My mum was a pre-school teacher so she's very experienced at dealing with small children, like nearly 40 years experience with more than 20 children a year. One bit of advice is to offer a child a choice of alternatives which you find acceptable. For example you don't ask "Do you want to go to school today?" because "No" is not an acceptable answer. But you might ask, "Which shirt do you want to wear to school today, this one, or this one?"

Similarly if someone is going to be a passenger it might be good to offer them some choice, some "agency", some deference (by letting them navigate if they want to,