"What would happen"?
Nothing different - they would STILL be Horcruxes and still work. The point of having special meaningful objects as Horcruxes was all about Voldemort's psyche, not magical laws.
Harry and Dumbledore discuss this in Half-Blood Prince, with Harry having a similar idea to the pebble one you mentioned (and for a far more interesting approach, I highly recommend the famous "Harry Potter and Methods of Rationality").
However, we should not congratulate ourselves too heartily. You destroyed the diary and I the ring, but if we are right in our theory of a seven-part soul, four Horcruxes remain.'
'And they could be anything?' said Harry. 'They could be oh, in tin cans or, I dunno, empty potion bottles...'
'You are thinking of Portkeys, Harry, which must be ordinary objects, easy to overlook. But would Lord Voldemort use tin cans or old potion bottles to guard his own precious soul? You are forgetting what I have showed you. Lord Voldemort liked to collect trophies, and he preferred objects with a powerful magical history His pride, his belief in his own superiority, his determination to carve for himself a startling place in magical history; these things, suggest to me that Voldemort would have chosen his Horcruxes with some care, favouring objects worthy of the honour.'
'The diary wasn't that special.'
'The diary, as you have said yourself, was proof that he was the heir of Slytherin. I am sure that Voldemort considered it of stupendous importance.'
'So, the other Horcruxes?' said Harry. 'Do you think you know what they are, sir?'
'I can only guess,' said Dumbledore. 'For the reasons I have already given, I believe that Lord Voldemort would prefer objects that, in themselves, have a certain grandeur. I have therefore trawled back through Voldemort's past to see if I can find evidence that such artefacts have disappeared around him.'