what is the cheapest travel from biaritzz to paris

Here's the website for the French rail company: SNCF Connect

Fares for May 14 (a Sunday) are nearly all over 100 euros. It would be considerably cheaper to travel on May 15 if you bought the tickets now.

Skyscanner.com is the website I use to check schedules for intra-European flights. It will give you a good idea of the fares, but you must go to the website of the airline to be sure you have accurate fare info. Skyscanner lists an EasyJet departure at 3:25 PM for just 36 euros. However, that doesn't include seat selection, checked baggage, etc., so you'd need to check all the add-on fees on the EasyJet website.

Although you might be able to travel by bus for less money, it looks as if the trip would take about 12 hours.

https://community.ricksteves.com/travel-forum/france/train-travel-from-bordeaux-to-paris-26c80e93-7bf4-4a7d-991d-72da2f746303

Where are you really originating? How many travelers?

A quick looks reveals air and train fares are about the same. (~$120- $140 +/-) If you have 4-5 travelers a rental car may be the cheaper choice. ERAC has an AT 7 passenger van for ~$250 for a one day rental, one way. Plus gas and tolls, so 3 people is close to a push, 4 or 5 makes it cheaper

I would not recommend driving, it is a long day and gas and tolls would come close to $200 for this 500-mile journey.

However, you could look into splitting your train ticket between a Biarritz-Bordeaux ticket and a Bordeaux-Paris (or Massy, a Paris suburb, for some low-cost Ouigo trains) ticket, allowing plenty of time (>1 hour, and perhaps even 90 minutes if the Bordeaux-Paris train is a low-cost "Ouigo") between the two.

By doing so, I am finding combinations that cost 70€ or so instead of close to 100€.

If you do end up in Massy TGV station instead of Montparnasse, you are 30-40 minutes away from central Paris on RER B and the ticket is cheap, so it is not a big deal.

EDIT - I hadn't checked prices for direct Biarritz-Paris trains. If they're 87€ rather than 100€, then in the context of a vacation in Europe it is definitely not worth changing trains in Bordeaux to save 10-15€ or so per person.

Biarritz to Paris Montparnasse TGV tickets for that date start at 87€ for a direct ticket 2nd class, and it takes 4h13 minutes.

Indirect travel takes at least an hour longer, but can be 10€ cheaper. It depends how much an hour of your time is worth :)

look at your options on rome2rio.com

The cheapest is the bus with Blablacar.
Departure from Biarritz airport - Arrival at Paris Bercy.
Average price 35€

(travel time about 11 hours)