Does she have a concealed weapons license? Your OP implies she does not but is not clear.
If she does not have a license, there are different laws in every state she will travel. In Tennessee, Tenn. Code § 39-17-1308(a)(3)(C) allows a person to carry a handgun without a permit on his or her premises. A private hotel room should qualify under this provision. However, until a person is within his or her hotel room, a handgun would need to be unloaded and not carried concealed on or about the person.
Also be aware that when she gets to Ohio, there are very cumbersome rules specified in Ohio Rev. Stat.§ 2923.16 for possessing and transporting firearms in vehicles, violations of some of which are punishable as felonies. A person who does not have a concealed handgun license recognized in Ohio must transport the handgun unloaded and unload all magazines and speedloaders capable of being used to load the handgun.
If your mother has a concealed weapons license, all of this (except Ohio's car carry rules for CHL holders) is moot. If she does not have one, she needs to learn the different rules for each state and you should encourage her to get a license to avoid these problems in the future.