The correct answer is option 'c', stay.

Explanation:
- In this sentence, the subject is "my family and I" which is a plural noun phrase.
- When the subject is plural, we use the base form of the verb (also known as the infinitive form) without any additional 's' or 'es' at the end.
- The base form of the verb "stay" is "stay", which is why option 'c' is the correct answer.

Let's break down the options and see why only option 'c' is correct:

a) staying:
- This is the present participle form of the verb "stay".
- It is used to show an ongoing action or a state of being.
- However, in this sentence, we need a simple present tense verb to match the subject "my family and I". Therefore, option 'a' is not correct.

b) stays:
- This is the present simple tense form of the verb "stay" used for the third-person singular subject (he, she, it).
- Since the subject in this sentence is "my family and I", which is a plural noun phrase, we cannot use the third-person singular form of the verb. Therefore, option 'b' is not correct.

c) stay:
- This is the base form of the verb "stay".
- It is used for all plural subjects and the first-person singular subject (I).
- In this sentence, the subject is "my family and I", so we use the base form "stay". Therefore, option 'c' is correct.

d) stayer:
- This is not a verb form of the word "stay".
- It is a noun form that refers to a person who stays.
- Since we need a verb in the sentence, option 'd' is not correct.

In conclusion, the correct answer is option 'c', stay, because it matches the subject "my family and I" and is in the simple present tense.