What a horrible waste of time and money
AFAICT, the basis of this action is that the country's entry to the European Union happened through the European Communities Act 1972, and triggering Article 50 of the Lisbon treaty without an act of parliament would be using prerogative powers to override that legislation. Since the prerogative powers are generally subject to legislation, as the sovereign-in-parliament is sovereign, not the sovereign, then using them to override legislation in this way would be unlawful.
But. The Lisbon treaty was added to UK law by the European Union (Amendment) Act 2008. So doesn't that legislation incorporate the Lisbon treaty into UK law, giving the government the right to trigger article 50 when it wants?
I'd be interested to hear informed opinion counter to this position; as far as I can tell, existing legislation enables the government to trigger article 50 without a new authorisation from parliament.