A familiar contraction of Harry (for Henry). Similarly, Dol is a contraction of Dorothy; Mol, of Mary, etc.
The substitution of P for M as the initial letter of proper names is seen in such examples as Polly for Molly, Patty for Martha, Peggy for Margy (i.e. Margaret), etc. (See Elizabeth.)
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894