If you're considering spot varnish, don't choose an uncoated stock that absorbs the varnish and thus negates the effect you're trying to achieve. Likewise, if you're aiming for a vibrant, colorful piece, use a coated stock that allows the ink to sit on top of the paper instead of soaking into it. If you don't want the shiny look of coated paper but still want the vibrant colors, compromise with a matte-coated stock. If the piece has an extremely long or indefinite shelf life, an acid-free paper will resist deterioration and maintain its color for decades. Ordering special stock.
Stocks vary in price and availability.
The perfect stock for your piece may have to be ordered directly from the mill and only in certain quantities. The printer needs to have enough time to order and receive it. If that happens, ask that it be stored for your future use. Ask the printer for help. The earlier in the design stage you ask for help, the better.