你的上面几句翻译都不对：

他开会迟到了，谎说自己的车坏了。

误：He was late for the meeting and lied that his car broke down.（lie不及物，其后不接that-从句）

误：He was late for the meeting and lied about that his car broke down.（介词后面习惯上不接that-从句）

误：He was late for the meeting and lied about the fact that his car broke down.（在介词后接 the fact that，语法上是对的，但意思不对，lied about the fact that his car broke down 的意思是“对车坏这个事实说谎”，即“车坏”是事实，但他对此说谎，比如说成“车没坏”——不符合上面的中文意思）

可以考虑用 lie about 来翻译：

He was late for the meeting and lied about his car breaking down.

用英语表示“谎说……”常用的表达方式：

He told a lie and said that he hadn’t seen her. 他撒谎说他没见过她。

I told a lie when I said I liked her new haircut. 我撒谎说我喜欢她的新发型。

He lied to me saying that he didn't get my message. 他向我谎说没收到我的信息。

When she asked what I thought of the meal, I lied politely and said it was very good. 当她问我对饭菜有何想法时，我客气地撒谎说非常好。

He lied to me, saying his date was one of the girls in the show. 他对我撒谎说他约会的对象是参加演出的女孩儿其中的一个。

She lied to cover up for her husband, saying that he had been at home with her on the night of the crime. 她为了帮丈夫掩盖，撒谎说他在案发当晚一直在家陪她。

除了用 lie 外，还有其他一些方式，视句型和句意需要而定，比如 make up some story about doing sth 就是其中比较常用的一个：

I'd make up some story about having to go home to see my sick. 我会谎说我要回家看生病的母亲。

He made up some story about having to be at his aunt’s wedding anniversary. 他撒谎说他必须去参加他姑妈的结婚纪念。