What’s the best you could do? This was the first album I ever got, I’d mean a lot to own it. 是什麼意思

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It sounds like the person wants to buy an album.

In this context, I would interpret "What's the best you can do" as "What's the lowest price you could give me, what's the lowest price you could sell the album for?"

"It'd mean a lot to own it."
This is a little hard for me to explain. It means it they really value the item, and owning it would be very special for them.

This picture means a lot to me = This picture is very special to me, they really value the picture.