a. 对于 (Duìyú) = to; for; about; with regard to; towards.
对于 (Duìyú) usually emphasizes the thing that is being pointed to. 对于 (Duìyú) is a preposition. It looks towards the object referred to, and usually the person or object referred to has adopted a certain attitude or has come into some kind of situation.
Duìyú diànnǎo, wǒ shénme dōu bùdǒng.
对于 电脑， 我 什么 都 不懂。
I don’t understand anything about computers.
b. 由于 (Yóuyú) = “due to” or “owing to.”
由于 (Yóuyú) + Cause or Effect.
c. 至于 (Zhìyú)= as for; as to; concerning; with regards to.
至于 (Zhìyú) is used as a preposition, it points to another topic and emphasizes changing the subject of conversation or emphasizes a degree.
Zhè diǎn xiǎo shìqíng, zhìyú zhème shēngqì ma?
这 点 小 事情， 至于 这么 生气 吗？
Why are you so angry over such a small thing?
d. 关于 (Guānyú) = about; on; with regard to; with respect to.
关于 (Guānyú) is a preposition which emphasizes a range, and it introduces all things that are related to it. Usually, 关于 (guānyú) is used in a formal way.
Guānyú zhè jiàn shì, nǐ néng gěi wǒ yì xiē jiànyì ma?
关于 这 件 事，你 能 给 我 一 些 建议 吗？
Can you provide any suggestions on this matter?
In this case, 生病 (shēngbìng) is the cause of the effect/result “晚来了十几天 (wǎn lái le shí jǐ tiān.).” Therefore, B is the correct answer.
Yóuyú shēngbìng, tā wǎn lái le shí jǐ tiān.
由于 生病， 她 晚 来 了 十 几 天。
Because of the illness, she came ten days later.