Fast Train from Paris to London for a day

Morris, you book the Eurostar. At this date the tickets will not be as cheap as they were a couple of months ago, but cheaper then buying any later, so buy now. Second class is fine. You are allowed to bring food and drinks onboard. Get earliestrrain I can and later one home.
Buy from Eurostar. com .
Journey is 2 1/2 hra city center to city center

Yes. If you want to spend at least 7 hours door to door for a day trip. Not me.

The Eurostar will work just fine. Just be aware that you need to be checked in 30 minutes before the train departure.

If they're staying in Paris for a month, it makes much more sense to go to London for at least a few days. Since the train is relatively expensive, they will get more value for a longer stay.

And I'll just emphasize that if they want affordable tickets on the Eurostar, they should book as soon as they know their plans. The prices can get stratospheric at the last minute (see this link for one poster who found out the hard way): https://community.ricksteves.com/travel-forum/transportation/london-to-paris-a9cfe0aa-2fc4-48a8-8754-26abaf3c2ee5

I would also prefer to stay over a couple of nights in London, if time allows. They should look at Eurostar tickets prices now and lock in anything they are ready to commit to. Prices should be the same through Eurostar.com or our US link and both sell these as e-tickets. Flying can also be cheap, but takes longer after you add in transport to/from the airports, which also aren't free.

We did the one day trip from Paris to London and loved it. We took the first Eurostar out in the morning and the last train back at night. We booked 90 days out and got the best price on tickets. Second class is fine. We were in a Paris apartment for 30 days. We considered spending a night in London, but ruled it out. The thing to consider if you choose to spend the night is you will be lugging a bag onto the train and will waste time checking into and out of the hotel. There is something wonderful about hopping on a train in Paris, carrying only a purse, and then arriving in another country free as a bird to just enjoy every minute of the day. I would do it again in a minute!