Post house holtel.
Take your pick from top hotels to b&b's. It depends what your budget is and how you like your comforts. Cruise.co.uk would probably be able to book a 3-5 star hotel for you near to the docks. Or if you'd rather stay in a b&b try looking on http://www.bedandbreakfasts.co.u.
The De Vere hotel is the only 5 star in Southampton or there is the 4 star Holiday Inn opposite the De Vere & close to the port. Cheaper options are the Novotel & Ibis.
Me and the good lady stay in the Premier Travel Inn Southampton West. £51 + £15 for breakfast.10 Min from the port 2 min fron the M Way. Nice clean and good grub to start your day off.