【1】He has a mother who likes watching TV.【正确】【a mother这里a不表示数量。】

备注：以上侧重点说明“妈妈拥有某种爱好”相当于是His mother likes watching TV。或者 His mother’s hobby is watching TV.而不是说明有1个....的妈妈

【2】He has a mother, who likes watching TV.【正确】【a mother这里a表示数量。】

备注：以上侧重点说明“有一个妈妈”然后补充说明妈妈有某种爱好。相当于He has one mother and her hobby is watching TV. He has one mother and she likes watching TV.

以下2个句子和以上2个句子有所不同【按逻辑和常理，妈妈只能拥有1个，但儿子可以有1个或者1个以上。】

【1】The old man has a son who is a doctor.【限制性定语从句表示：不止1个儿子】

备注：以上侧重点说明：儿子不止1个，其中1个为医生。相当于The old man has three sons and one of them is a doctor.

【2】The old man has a son ，who is a doctor.【非限制性定语从句表示：仅仅只有1个儿子】

备注：以上侧重点说明：儿子仅仅只有1个，这个儿子是1名医生。相当于 The old man has only one son and he is a doctor.

以下2个句子和网友提供英语句子定语从句用法完全一样

【1】Tom has a mother who is beautiful.【定语从句起修饰/描绘的作用】

=Tom has a beautiful mother.

【2】Mary has a face which is beautiful .【定语从句起修饰/描绘的作用】

=Mary has a beautiful face.

几位老师说的都有道理，我比较认同大鹏老师说的分析。

限制性关系分句除了表示“区分性”，也表示“描绘性”。

一个简单的思路是，对于逻辑上只有“一个”的人或物，其后都可以接描绘性的限定性关系分词，比如：

have a head that / which...

have a face that / which...

have a nose that / which...

have a stomach that / which...

have a father that / who...

have a mother that / who...

……

顺着这个思路，顺着这些用法完全可以查到不少权威实例！

赞同几位老师的解答，我仅补充几个外刊例句，供大家比较体会。我理解，语境决定具体含义。

1.

2. 

3. 

4.

5.

我觉得网友的句没有问题，我同意蒋老师的分析——这里的不定冠词表示“身份”，而不是表示“数量”。这里的 a mother who...是指一个什么样的母亲，而不是强调一个或两个母亲。

我曾在张道真的《现代英语用法词典》上记录了这样一个例句：

Although she had a husband who loved her, she felt her life was lacking in romance. (Chambers)

这里的 she had a husband who loved her 与网友句子中的 He has a mother who likes watching TV 用法是一样的。这里的 a husband 并不意味着她有多个丈夫。

如果将网友的句子改为非限制性定语从句，则表明后面的定语从句只是一个信息补充（可有可无），单独说 He has a mother 则可能需要特定的语境，比如：He has no father, no brother and no sister, but he has a mother, who loves him very much / who likes watching TV.

He has a mother who likes watching TV.

句子没有问题。句意：

他有一个爱看电视的的母亲。