My father, Richard Haller, played guitar and sang a lot when I was growing up, generally a mix of folk songs and other material that he had heard and liked. Some of those songs have made their way into my own repetoire (Milk Cow Blues, Sister Kate, Little Red Riding Hood to name a few), so clearly he was a big influence on me. He and my mother will also have music parties from time to time when family and friends would come over and play and sing, which is something that I've tried to do myself from time to time. My cousin David has a very characteristic style, whether playing on guitar or piano, and he's also been an inspiration.
Looking outside of the family ... my guitar playing tends to be pretty rudimentary, and I'm not sure how much resemblance it has to any of the people I listen to. I have a few 'show' pieces which are more elaborate and which are based on specific songs ... I play a guitar version of Robert Johnson's 'Kind Hearted Blues', the Dave Van Ronk version of 'That'll Never Happen No More', and Tommy Johnson's 'Big Road Blues'. Beyond that, I've listened to a lot of Johnny Cash, Mississippi John Hurt, Dave Van Ronk, Lightnin' Hopkins, David Rawlings/Gillian Welch, and Paul Geremia.
For my piano playing, a big influence was Jimmy Yancey, who tended to play a lot of slow-to-mid-tempo blues songs with some pretty spare piano playing, which was attractive to me because I've never been much of a speed demon on the piano. His playing, especially in his later years, tends to both be pretty simple and very beautiful. If you have a copy of my CD, the first song on it, 'Eleven Past Two', is my attempt to either evoke or rip off Jimmy Yancey. (This includes the title -- Jimmy Yancey had blues songs titled 'Two O'Clock Blues', 'Four O'Clock Blues', and 'Eternal Blues'.)
I've also listened to a lot of Thelonious Monk, although I'm not sure how much of his style I've managed to incorporate ... I've always felt that there was a tremendous amount of information for a blues pianist packed inside his Blue Note recording of 'Misterioso' if I could just unpack it. Other pianists I've listened to a lot have been Professor Longhair and Otis Spann and maybe Fats Domino. I've never really gotten into rock piano (except for Little Richard, but his playing is generally too fast for me), which is reflected by the fact that virtually all the things I play on the piano are blues or blues-based.
I have trouble pinning down general influences on my songwriting, but different songs I've written have been in different styles (although I'm not sure how clear this is to people who aren't me). The general attitude probably owes a certain amount to They Might Be Giants and Randy Newman and probably Weird Al. ('Tea In My Teacup' in particular strikes me as a Randy Newman-ish song.) Some song ideas come to me out of the blue (in a dream, for instance), while others are pretty conceptual (the idea for 'Lady Luck' was to write a song about a guy who has been mistreated by women, which is a common enough song theme, with the twist that he didn't really mind or let it get him down; 'Hybrid Car' came out of the thought that as far as I knew nobody had written a song that tried to make hybrid cars sexy, and then combined with my general amusement of songs that talk about how awesome the singer is). It's tempting to list Andy Breckman as an influence, but I only started listening to him pretty recently, after pretty much all of the songs that are stylistically similar to his had already been written ...