Renting DVC points for a night or two would most certainly be the cheapest option but as PP noted, people typically book 7-11 months in advance for DVC. There may not be availability at this point or you may need to be flexible with your dates or which night during your trip you stay at Aulani.
If it were me and budget was the most important concern, I would stay elsewhere on the island like Waikiki and just visit Aulani for lunch or dinner or even a spa treatment. You could also use the beach and lagoon at Aulani, as all beaches in Hawaii are public. You just would not be permitted to use Aulani's chairs and umbrellas, but you could bring your own beach chairs or blankets and hang out on the beach and enjoy the lagoon at no cost. The only thing you'd be gaining by actually spending one night at Aulani is experiencing the room and the pools. While they are both very nice, I don't know if they are worth the cost/hassle of staying just one night at Aulani. The DVC studio rooms are smaller than a standard hotel room and if you're able to book a standard one, which is cheapest, the view will be nothing to write home about. And the pools are great but its not hard to find great pools at other resorts too.