For trip planning, start here: https://www.londontoolkit.com/
There are many great hotels in London. Premier Inns are noted for being moderately priced and with good standards.
Some specific advice from CC concerning how to get from Southampton to London:
Our ship’s berth is 101 (City Terminal Dock)
The 07.30 bus would be far too risky. Book the 08.40 at £12, or the 09.50 at half that fare. Journey time 2.5hrs
Best to book reasonably far in advance, they do sell-out.
Book from Southampton coach station to London Victoria coach station.
Do check their luggage limits if you plan to bring a mountain of luggage.
Train schedule is out to part-way through May, prospects are that the rest of the month will be exactly the same (but for research do choose the same day-of-the-week)
Three to four direct trains per hour. so with luggage, it’s easiest to ignore those which involve a change of train.
No need to pre-book, but I don’t know if its a little more expensive to buy at the station.
Bear in mind that trains at that time in the morning will be busy with commuters and business people heading to London, that can be no fun with luggage or if you have to stand.
At £41 it’s twice the cost of the bus but gets you there in 90 minutes.
There are no cheap mega train tickets to London until gone noon, and the saving won’t be worth that wasted time.
Your journey is from Southampton central station to London Waterloo. There’s also an hourly direct service to London Victoria train station, but it takes a longer route, journey time about 2.5 hours.
Check those trains closer to the time – there’s been talk of a rival operator for the Southampton – Waterloo route, which might bring down the prices.
Both bus and train stations are a 15 min walk or a £6 taxi ride from City Cruise Terminal.
Bus or train, each to their own.
A Google map around Westminster Abbey and Churchill War Rooms