I'd highly recommend staying on one of the ones on Stour Road, the train station is only a two minute walk away and then so is the high street with all the shops. Make sure you book in advance though as they usually fill up really fast!
I stayed in the Stour Villa which was fine, didn't have wifi though but I did manage to borrow an iron.
If money isn't an issue and you aren't driving you could always stay in one of the hotels in Southampton and then get the train in the morning straight to Christchurch and a taxi to the airport. I stayed in the Ibis which was next door to the station for about £60, very nice room, left me very prepared for the sim.