Great question. I think that the character of Harrison Bergeron would be perfectly able to lead a successful rebellion agianst the society depicted in this story if he were to use different methods. The problem with Harrison Bergeron is not that he is deficient in any of the characteristics necessary to launch a successful uprising, but that he chooses to make a stand on public television in a way that makes it easy for Diana Moon Glampers to end his rebellion almost as soon as it is started.
From what we are told about Harrison Bergeron, we can see that he is incredibly intelligent, he is attractive and very charismatic. Note the way that when he bursts into the television studio, he easily compels others to obey him and to join him in his revolt. This leads me to infer that if he had been more subtle and cautious about his rebellion, working slowly to set up a rebel movement, he could have successfully overthrown the regime. However, his desire to act alone and in a public way makes his rebellion worthless and easily controlled, as Diana Moon Glampers shows.
If Harrison Bergeron were less impulsive and he mapped out his rebellion better by enlisting others in his cause, it seems reasonable that he perhaps could have gained control of the government.
One indicator that Harrison Bergeron would have been capable of enlisting others in a rebellion is the ease with which he manipulates people mentally and physically. For instance, when he suddenly appears at the television station declaring himself the Emperor and immediately selects his "empress," her willingness to comply is equally swift. Then, when the musicians do not obey his orders to "Play your best," and Harrison "snatche[s] two musicians from their chairs," waving them around with ease as though they are the batons of a conductor, every one of the other musicians quickly becomes "much improved." With alacrity they comply with Harrison and play as he wishes. So, with this type of influence and control over others, Harrison again demonstrates the potential to effect a successful rebellion.
Clearly, Harrison possesses superhuman abilities. While with his empress he is able to "abandon" the laws of gravity and motion as he dances with her, he is also able to "neutralize gravity with love and pure will." His strong will makes Harrison a force that could be capable of leading a nation.