On the one hand, I suggest there is nothing significant about the disciples' question. They were obviously fascinated with Jesus' teaching. Consequently, they wanted to spend more time with him.
Put yourself in their position. You're talking to a respected teacher on the street, and you are fascinated by what he has to say. When he turns to walk away--since you want to hear more, you simply ask him:
"Uh, Rabbi, where do you hang your hat? We'd love to hear more from you, if that's all right with you."
Contrary to popular thinking, Jesus did have places to stay, whether in Nazareth, Capernaum, Bethany, and probably other places that escape my thinking. His base of operations, so to speak, was likely a house in Capernaum. Only late in His public ministry did He say,
"The foxes have holes, and the birds have nests, but the Son of Man has nowhere to lay His head" (Matthew 8:20: Luke 9:58).
Even that expression doesn't necessarily mean Jesus had no place to crash; it does mean He owned neither house nor land; in other words, things which we mortals consider to be relatively permanent and immoveable, were not of interest to Jesus. He had bigger fish to fry, so to speak!
On the other hand, there is something significant about the disciples' questions about "where are you going?" and"where are you staying [, Jesus]?" We read in the Tanakh,
"The earth is the LORD'S, and the fullness thereof; the world, and they that dwell therein" (Psalm 24:1 KJV).
Jesus, as the eternal Word, the Son of the living God, owned everything, since
"All things came into being through Him, and apart from Him nothing came into being that has come into being" (John 1:1-3).
Given His role in creation, Jesus' humble self-emptying is all the more amazing (see Philippians 2:5-7, especially v.7). Clearly, He shunned five-star accommodations--though He was certainly entitled to them--in favor of an itinerant ministry with its coach seating, the occasional crashing on a friend's floor (or mat), or even sleeping under the stars!
I used to sing in Sunday school:
He owns the cattle on a thousand hills,
The wealth in every mine.
He owns the rivers and the rocks and rills,
The sun and stars that shine.
Wonderful riches, more than tongue can tell,
He is my Father, so they're mine as well.
He owns the cattle on a thousand hills,
I know that He will care for me!
Again, the Creator and Sustainer of the universe was under no obligation to become poor for the sake of us sinners. By rights, he could have lived in the finest mansion or palace imaginable. We are told very clearly in Scripture, however, that
"You know the grace of our Lord Jesus, that though He was rich, yet for our sakes He became poor, so that we through His poverty might become rich" (2 Corinthians 8:9).
Sorry for the prolix answer, but I sometimes (OK, almost always) get on a roll and stray from the question.