0. the interviewer has discovered that her co-workers and a neighbour are (what? ) collectors.

1. the speaker, gary brockman, collects buttons of (what period? ) .

2. while in new york, the speaker used to work in (what field? ) .

3. the workshop was held for (whom? ) .

4. among the workshop participants there were (how many? ) button collectors.

5. the speaker wanted to talk to the button collectors (when? ) the workshop.

6. button collecting became an organised activity in the usa (when? ) .

7. the activity of the organisation was temporarily stopped by (what? ) .

8. (what? ) in different classifications of buttons are held at the national button show.

9. buttons can give an insight into our culture and (what else? ) .

10. buttons help trace the development of artistic (what? ) .

ОБА варианта верны.
1) Mother`s bag- здесь использован притяжательный падеж. Существительное Mother`s стоит в притяжательном падеже, отвечает на вопрос чей?чья?=сумка чья?=мамы=мамина сумка
Необходимо помнить, что существует ещё конструкция с предлогом of, которую можно использовать ВМЕСТО притяжательного падежа. Она полностью заменяет существительное в притяжательном падеже. Данный предлог соответствует родительному падежу в русском языке и отвечает на вопрос кого? чего?
2) Bag of mother=сумка кого?=мамы=мамина сумка

Дополнение к ответу: с предлогом of употребляются ЛЮБЫЕ существительные(и одуш. и неодуш)

Newton's great work is called "Principia" was published in 1687. More than a century ago the Russian chemical society was organazied and named after Mendeleev. The subjects in the first and second years of the students studies are very important for their future specialities. The government wasn't interested in the invention which popov made ( was made by Popov).

1) We were told about the film which was made several years before. A very interesting article which is published in the magazine is available in our library. The instrument is made of the new material. We don't know much about this subject. We heard so much about the cosmonauts who came to our town. Have you seen the new device's main components it consists of?