I’m thinking of renting a car for a day from San Sebastian in order to drive around French Basque Country. In addition to the basic car rental cost, are there any other items I need to be concerned about, especially since we’ll be driving between Spain to France?
No. It's really easy and no concerns whatsoever. As long as you inform the rental agency that you plan to go to France, which will usually involve an extra fee.
I'll add that if there is an extra fee, it will be nominal, and last time I did this the other way around (Biarritz to Spain and back) there was no fee (Enterprise).
We just rented a car in Bilbao from Sixt and drove into France. When renting there was a question like "what countries will you be driving in" or something like that. I believe it added a fee of approximately $35.00.