建议改为：I met John yesterdy. To my astonishment, that strong, old friend of mine fell ill. 

▲名词性物主代词（mine, ours, yours, his, hers, theirs等）作定语修饰名词时，要使用of所有格形式。例如：

1. 泛指“我的一个朋友”时，说：a friend of mine

2. 特指“我的这个朋友”时，说：this friend of mine；特指“我的那个朋友”时，说：that friend of mine

注：这类表达往往带有感情色彩。

但是，没有见到过the friend of mine的说法，可能特指的程度远远不够。

3. 泛指复数名词时，说：

I decided to gam some friends of mine this Sunday. 这星期天我决定去找我的几位朋友聊一聊。

来自《现代英汉综合大词典》

two friends of mine：我的两个朋友。

4. 特指复数名词时，把this改为these，把that改为those。例如：

I shall have words with these stupid friends of mine!  我真应该跟我这些愚蠢的朋友理论理论！

来自柯林斯例句

注：这类表达往往带有感情色彩。

这道改错题，对于中学生来说，是有点太苛求了。不管什么目的，命题人的思想是有问题的。

▲以下是部分例句，可参考：

I can't even remember the name of that old friend of mine. 我甚至连那位老朋友的名字都不记得了。

This friend of mine loves to drink tea. 我的这个朋友喜欢喝茶。

Then I remembered playing with this friend of mine and now I knew I fell off the tree.

然后我想起自己和朋友玩耍。 这时我明白了，自己从树上摔下来了。

I continued to walk on but this particular friend of mine rushed back to the guy to put him back on his feet again. 我继续行走，但是我那个特别的朋友返回去扶起了他。

This new book of mine is not up to my last. 我的这本新书不及我的前一本。

Where's that good-for-nothing son of yours?  你那个废物儿子在哪儿呢?

I wish you'd talk some sense into that son of yours.  我希望你能和你的儿子讲讲理。

We strongly protest against this remark of yours.  我们强烈抗议你说的这句话。

来自辞典例句

This proposal of yours hardly meets the case.  你的这项提议不甚恰当。

来自辞典例句

He impersonated this friend of yours in order to come to your party.  他为了参加你的酒会而冒充你的这位朋友。

来自辞典例句

That brother of yours is full of boasts.  你那个宝贝兄弟满口大话。

来自辞典例句

Apart from a few grammatical mistakes, this composition of yours is well written. 除了几处语法错误之外, 你这篇作文写得不错。

来自《现代汉英综合大词典》

When are you going to publish this work of yours?  你什么时候出版你这部作品?

来自辞典例句

There is no sense in fastening on that pretext of yours.  一味强调你那个借口是没有用处的。

来自辞典例句

There's a law in the land still, and you better control that tongue of yours.  这个国家还有王法呢, 你最好管住你那张嘴巴。

来自辞典例句

As for this Palace of yours, it gives me the creep.  至于你的这座王宫, 它可叫我心惊肉跳。

来自辞典例句

This plan of yours cuts into our schedule rather badly.  你们的这个计划要占去我日程表里不少时间。

来自辞典例句

This box of yours weights a ton.  你这只箱子真沉重。

来自辞典例句

"If I were you," she adds by way of some unsolicited advice, "I'd watch out for that girl of yours."

“如果我是你，”她主动建议说，“我会密切注意自己的女儿。”

来自柯林斯例句