Oh, there are of course a lot of possibilities before open hostility.
a) the most "easy" way (but since it would mean both sides act honestly and follow the logic, it is not very likely) - share the money which must be spend, share the cost of protect the mine AND share the profit.
b) bribe your superior liege lord (or the king) so that he give you the right to exploit the mine, and counteer actions of the other house...
c) look if you could claim the land from your neighbour before he even notice how much of worth it is. If there is something your neighbour wan't, than you could offer it him in exchange for a strip of land, "accidently" with the land you are after. Of course here you must be good in negotiation, so that the neighbour did not notice after what you are in fact. And if he later learns that he was tricked...well every good familiy feud must have a start...
d) in some cases (if the land is very densly populated) you may even try to exploit the deposit in the hope your neighbour did not notice it. You may try to disguise what kind of ore the mine is producing (if the silver is mixed with other ore than your neighbour may be not rush to build his own mine if he notice yours. And you could secretly dig into his area. That would mean you must disguise how important the mine os and how much it cost (spreading rumour of ore of poor quality, unstable ground, a haunted mine... *g*)
e) try to clear the right of who could exploit it by a fight of seven against seven to sho what the Smith wants
And so on and so on...