The way I saw it, they chose not to ellect a new one, but to let Sarutobi back on the spot. But it's up for interpretation. It's just a way of seing things. When they were going to choose Kakashi, a lot of people had to vote, both from konoha and from the fire country. I just assumed that when minato died, the same thing happened, but instead of picking a new one, they chose to let Sarutobi back on the spot. That's what I assumed anyway. But it's just an interpretation... :S
It's not an intrepretation; that's how it actually happened.
Hashirama Senju (First Hokage) >>> passes on position to Tobirama Senju (Second Hokage)
Second Hokage >>> passes on position to Hiruzen Sarutobi (Third Hokage)
Third Hokage >>> passes on position to Minato Namikaze (Fourth Hokage)
Fourth Hokage >>> dies without choosing a successor; Sarutobi reclaims the position
Third Hokage >>> dies without choosing a successor; Tsunade Senju (Fifth Hokage) takes position of Hokage
Fifth Hokage >>> defends villagers from Pain's Shinra Tensei; village is forced to choose a new Hokage; Danzo Shimura becomes candidate (but is not inaugurated)
"Sixth Hokage" >>> dies without choosing a successor; Fifth Hokage retakes mantle
Fifth Hokage >>> ? (no successor chosen as of yet)