Here are some suggestions
Youth Hostel
Genoa Hostel Passo Giovanni Costanzi 10, 16135 Genova www.ostellogenova.it
Located on the heights, on one of the most scenic points in Genoa, and very well connected to the rest of the city by public transportation.
Cheap, clean and decorous.
B&B
B&B La Meridiana within the Historical centre http://www.bblameridiana.it
B&B La Maddalena located in via della Maddalena, it's wonderful! http://www.lamaddalena.genova.it/
B&B located only in the Historical centre: http://www.irolli.it/bed_and_breakfast_genova.html
B&B in Genoa: http://www.bed-and-breakfast.it/citta.cfm?citta=Genova&IDregione=8
Camping
Villa Doria Campsite http://www.camping.it/liguria/villadoria/
Situated in a quiet and picturesque park area in Pegli, just west of Genova.
Thanks to Navebus, the new Genoa transport service, Porto Antico can be reached in 30 minutes.
Hotels
Genoa Hotelkeepers Association http://www.hotelsgenova.it/
Sometimes you can find interesting special offers...
Search Engines
Comparior http://www.comparior.com/ metasearch engine that compares B&B and hotel room rates all over the world giving you the chance to save time and money!
Comparior users, in case they decide to book one of the services listed in the page of results, are re-direct to the supplier’s website in order to complete the transaction.
Booking http://www.booking.com/
Booking offers highly competitive hotel rates in both cities and regional destinations – ranging from small family hotels through to luxury hotels.
Reservation service is free of charge. No booking fees, administration fees, or cancellation fees.
And last, but not least, the evergreen:
Venere www.venere.com
Expedia www.expedia.it