Pakenham Arms - Pakenham Street, London, UK - Pictorial Pub Signs on Waymarking.com

A quick browse of the internet indicates that there are a few historical, military Pakenhams of both Navy and Army backgrounds. As the pub sign shows a more land based scene, rather than seascape, it is assumed that this pub is named after Major General Edward Pakenham, brother-in-law of Wellington.

The Almost Chosen People website has this to say about him:
"The battle of New Orleans is famous for launching Andrew Jackson on his eventual road to the White House. Almost forgotten today is Major General Edward Pakenham, the British Commander. A brother in law of the Duke of Wellington, Pakenham had a distinguished career in the British Army, earning rapid promotion for his good service in the Peninsular War in Portugal and Spain.

At the battle of New Orleans Pakenham fell victim to American grape shot. An initial blast killed his horse and shattered his left knee. As he was helped to his feet, he was wounded in his right arm. After he mounted a horse, yet more grape shot ripped through his spine giving him a mortal wound. He was 36 years old.

His brother-in-law, Wellington, was bitter about his death:

We have but one consolation, that he fell as he lived, in the honourable discharge of his duty and distinguished as a soldier and a man. I cannot but regret that he was ever employed on such a service or with such a colleague. The expedition to New Orleans originated with that colleague… The Americans were prepared with an army in a fortified position which still would have been carried, if the duties of others, that is of the Admiral (Sir Alexander Cochrane), had been as well performed as that of he whom we now lament.

Pakenham’s body was placed in a cask of rum and shipped to his home in Ireland for burial. Pakenham was noted for having a surly temper in his life, and one of his relatives waggishly wrote: “The General has returned home in better spirits than he left."

Source Almost Chosen People blog.