Yes. I prefer "that". Technically, "who" should be "whom".
Do you have any friends who/that you'd be willing to stay with for the rest of your life?
Are both okay here?
I would only use 'whom' after 'with' (after a preposition - as many people do today).
"Do you have any friends with whom you'd be willing to stay for the rest of your life?" or
"Do you have any friends who you'd be willing to stay with for the rest of your life?"
That might be Odessa Dawn's implied question, and it's the non-technical answer for many people.
When you say "stay with", do you mean "live with" or "remain friends with"?
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.
Batman, are you aware that you could omit 'who/that' in that question without altering the meaning?
I wasn't aware of that. Thank you for letting me know!