No, the Secret Service is well-trained to protect the first lady of the United States and it would be unwise for her to forego this security. Carrying a gun would not be enough to screen the FLOTUS from the many dangers in today's society. Dedicated members of the Secret Service should ensure her safety.
The Secret Service was not nearly as worried about assasintions during Elenor Rooevelt's time as it is not. At that time, the only president who had been assinated was Abraham Lincoln. In the next few years, JFK would be assantited, Regan would have an attempt on his life, and Bobby Kennedy would be assinated.
No, the first lady today would not be able to carry a gun instead of accepting Secret Service protection, as Eleanor Roosevelt did. At the time of Roosevelt, carrying a gun was a standard practice for the general public--including the "bad guys." Today the threats come from more varied sources and the Secret Service is needed for intelligence as well as physical protection.
The first lady should not be distracted with her own personal protection and should not carry a gun. There needs to be adequate Secret Service protection at her side for all public events. The carrying of a gun is an added risk, the gun could be taken from her and used against her.