Avondale is awesome, sad to move out next week (see the Houses for Rent section if you're interested in renting my landlord's house!), especially when you have an ass crack of dawn brief/flight. When I get back from next deployment I'm probably going to get an apartment in 5 Points in Riverside and if I get shore duty here I'm definitely going to buy in Avondale.
At the end of the day though where you live in Jax really comes to personal preference. Right now I'm just speaking to NAS Jax since obviously these commute factors really don't apply to those at Mayport, etc.
Beach- Pros- you're at the beach and the bar scene there. Cons- 45 min commute to base.
Southside- Pros- lots to do, lots of single JOs live there. Cons- not really a lot of character, have to drive pretty much everywhere.
Orange Park- Pros- more family centered, suburbia, better schools in Clay County. Cons- long drive to everywhere, suburbia
Riverside/Avondale- Pros- close to work, awesome area with awesome character, decent bar scene. Cons- if you're looking to buy the homes are old and require much upkeep, rent/housing costs expensive, odds are you won't be living near a bunch of other JOs