What does it mean to prepare an audition piece in the style of the show?
Option 1: Stick with the show's Broadway era
Li'l Abner is from the era of classic Broadway musicals - those that were written and premiered between 1920 and 1959, a period that includes the “golden age of musical theatre.” Broadway Musical Home has a great list of classic musicals.
Option 2: Go with the show's composer
The production staff can tell a lot from how you handle a song by the same composer as the show for which you are auditioning. Li'l Abner has music written by Gene de Paul, and the Internet Broadway Database has a (short) list of other shows featuring his music.
Option 3: Research the setting of the show
Choosing another song from a similar time period or with a similar setting can add variety to your options. However, you are less likely to find a convenient list. Li'l Abner is set in a backwoods town in the 1950's. An internet search for "musicals with a country feel" might bring up some possible song choices.
Option 4: Skip Broadway altogether
This is definitely a riskier choice, but sometimes you can find popular music that fits the feel of the show for which you are auditioning. Do your research! Make certain the song you choose fits for a specific reason; maybe it has a similar beat, tempo, or chord structure.
Hopefully this has helped! If you have any additional questions about your audition selection, feel free to contact me. I am happy to be a pre-audition resource.