I imagine it's not much different than different real world currencies; they'll all become relative to one another through a common conversion rate (ie. USD).
Plus somewhere in the middle someone will create a meta-cryptocurrency payment service and sign up vendors to accept any type of cryptocurrency while making a conversion fee.
Maybe the value comes from the network of miners and users of the altcoin.
If a new cryptocurrency becomes popular and steals miners and users from an old cryptocurrency, the value of the new coin will increase at the expense of the old coin.
But if the new cryptocurrency brings in new miners and users, like Dogecoin has, then the value of both old and new can grow.
Plus somewhere in the middle someone will create a meta-cryptocurrency payment service and sign up vendors to accept any type of cryptocurrency while making a conversion fee.