How much is my book worth?

The inscription, while pretty cool, is probably less important than the edition and, especially, the overall condition. Since it was inscribed in 1859, it's almost certainly a first edition, but I can't say for sure that it's a "true first edition," which means first edition and first printing. I believe it was first printed by Bradbury and Evans in London, so you should look for that information in the first pages. If it's a first edition and in good condition, then it's probably worth a decent amount of money, but I'm not an expert myself, so you should bring it to the nearest rare book dealer in you area for an appraisal and possibly a sale.