Nothing that I could find in canon/JKR info.
However, one would imagine if there were properties, Professor Slughorn would have tried to obtain them in HBP, which we don't see happening - if so, Dumbledore or Harry (under Felicis) would have used them as bait.
Out of universe:
As described by ancient Roman writer Lucan, phoenix ashes are a crucial ingredient in a mythological resurrection potion (src: "Kronzck, Allan Zola and Elizabeth Kronzck. The Sorcerer's Companion: A Guide to the Magical World of Harry Potter". 2nd ed. Broadway Books: New York, 2004, p209).