What is the value of an 1862 US 3 cent piece?

Look closely at the "2" in the date. It may be clearly struck, or there may be a shadow of a "1" underneath, as the result of a re-cut die. Numismedia lists the following approximate retail values as of 09/2013:

Clear "2":

Very worn condition - $30

Moderately worn - $52

Slightly worn - $62

Almost no wear - $169

Uncirculated - $216 to $588 depending on quality

"2" over "1":

Very worn - $32

Moderately worn - $55

Slightly worn - $66

Almost no wear - $191

Uncirculated - $288 to $790

DISCLAIMER: The retail values quoted are the best available as of the date shown, but may vary significantly due to changes in the precious metals market. Also the wholesale (buying) price of a coin will be less than the selling (retail) price. A reputable coin dealer will be able to give a more accurate valuation based on inspection of the coin.