You will be able to make a custom prompt using custom dialogs. The only thing you have to know is that the message string you give to bootbox doesn't have to be plain text. It can be HTML, so you can put your own prompt in a custom bootbox dialog.
What you are trying to do is this (using Bootbox 4.x):
bootbox.dialog({
message: "First name:<input type='text' id='first_name'>",
title: "Custom label",
buttons: {
main: {
label: "Save",
className: "btn-primary",
callback: function() {
console.log("Hi "+ $('#first_name').val());
}
}
}
});