We are a married couple, 55 and 60. We will be in Barcelona for 5 nights Oct. 25-30. We arrive by train from Avignon and leave by air. We'll be rolling our luggage and taking the subway to the aprtment. I have narrowed the apartment to rent down to these two:

1. http://www.barcelonabestlodge.com/index.htm
120 euros a night, near Ciutadella Park.
2. http://www.negre.us/apartments/Valencia/41/
89 euros a night, in Eixample.
Both are 2 bedroom, even though there are just 2 of us. Barcelonabestlodge does not require a deposit; Negre Apts. requires 25% deposit and non refundable 50 eu booking fee if canceled.

We plan to walk and take subways or buses to see the sights. We are not interested in clubs or bars. We are looking for a neighborhood with plenty of good inexpensive restaurants that we can walk to for dinner.
Which is the better location for a tourist base? Does anyone have any experience at either one? Is it likely there will be discounts on apartment rentals if I wait until closer to the trip due to the lack of tourists coming? Has anyone seen any discounting yet?

It looks like you have two excellent choices. The Eixample place is near Gracia (just to the north of the Diagonal), which is a quieter district with lots of small shops and restaurants. It's also nearer Sagrada Familia, Parc Guell and other Gaudi sights.
You say you are not interested in clubs or bars, but the other is within a comfortable walk of the Barri Gotic and El Born, which have great dining spots as well as the Cathedral and Santa Maria del Mar church, Picasso museum etc. And it's nearer the beach, if that's important.
There are some discounts out there, but they are generally available for stays closer to the reservation date. Late October shouldn't be particularly busy, but you'll have to decide if the insecurity of not knowing where you're staying is worth a 20% discount. Both the places seem to be good bargains. And picking one now will allow you to plan your visit better.

I've stayed in the area of the one you got in Eixample 2 weeks ago. There are restaurants and cafes everywhere, though it does not necessarily have the "vibe" that most people associate with Eixample. You will find more of that in the left part of Eixample, Pg de Gracia and beyond. But it's by no means a "dead" neighborhood.

Buses are often the better choice, especially when you live near the big boulevards where many lines run.

If your train arrives at Est. de Francia, you may want to take a taxi since there is no subway at that station. And a cab will cost you close to nothing to the place at the Parc, and probably less than 10 Euros to the one in Eixample.

Our train from Avignon comes into Barcelona Sants on Sunday 10/25 at 6:00 pm. It looks like there is a subway to a stop close to both of these without having to change lines. A taxi to either could be pricey. Not interested in the beach in late October-probably going to be rather cool.
Cowboy1968-where did you stay, hotel or apartment?

You are absolutely right. From Sants you can take the subway to the apartment in C/Valencia.

The other one near the Parc requires one change from L3 to L1 at Pl.Espanya. The greyish line (if you have color-coded subway map) that does a direct connection between Sants and Arc de Triomf (which would be closest subway for the apartment) is not the Metro or subway but the suburban trains Cercanias/Rodalies which leave from Sants railway station (i.e. where you will arrive) and not Sants subway station (which is a bit of a walk through underground tunnels).

I stayed at a hostal, sort of. Trust me, the place(s) you found is/are better

We stayed in the Negre apartment D in the Valencia building for the first week in May. The apartment was very nice and the managers were very pleasant and helpful. Our apartment looked out over the large open interior of the block, while the one you are considering looks out over C/Valencia. We enjoyed the "Rear Window" experience. The neighborhood had a comfortable urban feel. It was very easy to stroll and groceries and restaurants were nearby. Many of the modernista apartment buildings are nearby as is Sagrada Familia. We also liked taking the nearby city buses.

We also took the Metro from Sants to the closest stop -- Verdaguer. We found that there were a lot of stairs and no escalators, so be prepared for this where arriving with your suitcases.

We were also concerned about the cancellation fee, so we included it in our travel insurance.

Enjoy your time in Barcelona.

Thanks for all of the input. I'm going to go for the Negre apartment. The location appears good and sunny101's comments help, from someone who actually stayed there. I don't have travel insurance for this trip so I'll just have to hope I don't have to cancel the trip and lose 50 euros. Plus I like the fact that I can put the whole charge for the stay on a CC.