Try any of the dozens of Accor hotels - like Ibis or Sofitel, Novotel or ETAP, etc. - they will fill your bill - modern, clean near metro lines - www.accorhotels.com - I like the IBIS right outside the Gare du Lyon - TGV trains to Spain I believe would leave from there. Lots of shops nearby.

We have stayed at the Ibis near Bastille and it is exactly what Pal says. And their breakfast is very nice. Their staff was very helpful.
The rooms are very small and utilitarian, but they do have triples that would be perfect for your children and grandchild which would make it very attractive for your family.
You don't say what level of mortgage you don't want to take out!! There are many more possibilities also.

Trains to Barcelona would leave from Gare de Lyon, so something in that general area would be convenient for that purpose. I presume you are looking for two rooms, a double and a triple. Or, if not right near Gare de Lyon, something in the Bastille area could be good also (or the Marais if you go farther, but hotels are more expensive there, in general). I you travel lightly, you can get to Gare de Lyon from the Bastille metro stop in one stop, no transfers. I've done that and it was very convenient.

Here is a budget 2* hotel on bd Beaumarchais which is right near the Bastille metro. It has very reasonable rates and triple rooms. http://hotel-lyon-mulhouse-paris.com/ It even has AC which isn't common in a 2* hotel and could be nice in June, although not a necessity.

I think that area would be very convenient for everything, they have lots of cafes and restaurants nearby, a great Sunday morning open air market, and good transportation connections.

Trains to Barcelona would leave from Gare de Lyon,>

well unless OP is talking about the hotel train - night trains leave I believe from Gare d'Austerlitz, a short walk across a bridge over the Seine from the Gare de Lyon.

I agree with PalenQ, every time we could in French cities, we booked an Accor hotel. Novotels were my favourite, always clean, well equipped and kept to a high standard. For cheap and cheerful, the Ibis line also works.Booking a few weeks ahead will gain you a discount and they have various packages available too.

I second Christina. I stayed there. Very nice. Close to Bofingers too, as I recall. The big impact on me was there was a bakery "downstairs" and the aroma of baking bread when I woke up!

Thanks for the input. I will now begin my search.

Have you thought of renting an apartment?