Can I stay at a non-Muslim friend's house overnight?

Yes, usually. Staying overnight with a non-Muslim friend is allowed, especially with a friend of the same gender in a proper household. What matters is that you can keep your prayers, your modesty and your halal food, and that you are never alone overnight with someone of the opposite sex who is not your close relative.

The basic rule

Nothing in Islam prevents you from being a guest in a non-Muslim home. The key limit is about privacy with the opposite sex. The Prophet ﷺ said that no man should be alone with a woman unless a mahram, a close relative she can't marry, is present (Sahih al-Bukhari 5233). So:

A woman staying with a female friend in a home with other family members is fine, as long as she keeps modest around any men there.
A man staying at a female friend's home alone, or the other way round, is not allowed.
Married couples can stay together with a mixed-gender host couple, with normal modesty.

Sleeping and privacy

The Prophet ﷺ said that a man should not lie with another man under one covering, nor a woman with another woman (Sahih Muslim 338). In practice, this means separate beds, sofas or sleeping bags, and not sharing a blanket. Dress modestly for the evening and morning as well. Women who wear hijab should put it on before leaving their room if men of the household could be around.

Prayer, food and cleanliness

Prayer: Ask in advance for a quiet corner and say what time you wake for Fajr. Most hosts are happy to help. Use a phone app to find the qibla.
Food: Tell your host what you eat, or bring snacks. You can be a guest where others drink alcohol or eat pork, as long as you don't consume it and aren't pressured. See What if my roommate drinks alcohol?
Water for the toilet: Bring a small bottle if you wash after using the toilet.
Wudu: Mention that you will need to use the bathroom sink, to avoid surprises.

Being a thoughtful guest

Good manners are part of Islam. Bring a small gift, help clear the table, keep your things tidy and thank your host. A kind guest is a good way for friends to see what Muslims are like. See Can I keep my non-Muslim friends after becoming Muslim?

If you're travelling together, see Can I travel with non-Muslim friends as a Muslim? For children, see Can Muslim kids go to sleepovers?

When to say no

It is wise to skip or shorten the stay if the evening will centre on heavy drinking, if you would be alone with someone of the opposite sex, or if you can't pray or eat properly. A hotel or a short visit with a late taxi home is a polite alternative.

Key points

Staying with a non-Muslim friend is allowed in an appropriate setting.
Avoid being alone overnight with the opposite sex (Sahih al-Bukhari 5233).
Keep separate bedding and modest dress.
Plan for prayer, halal food and wudu beforehand.

Tip: Send a short message before you go: "I pray five times a day and eat halal, so I'll need a quiet corner and I'll bring my own snacks." Hosts nearly always appreciate knowing in advance.