Why would you think a travel router can magically improve the WiFi signals. A travel router will not get any better signals than your xbox. If you were talking a cell phone maybe because the antennas tend to be small but even those work pretty good.
Now if there is a ethernet jack in the room a travel router could be used to create a new strong wifi signal from that.
In many cases you will have issue using a router of any kind with the hotel systems. Most have some form of login page or other restrictions that make it very hard to connect to a hotel system via a wireless connection.