While beast druids during the year of the gryphon was already really strong, the year of the hydra offers some new fun mechanics ready to fill the gaps of rotated cards. Currently, I have about 67-70% winrate with this deck in about 15 games.

Compared to the former beast druid, this version is slightly slower but builds wide and strong boards nonetheless. The inclusion of Wildheart Guff and Ivus, the Forest Lord help for big turns in a slower game against control decks. Now that Warriors were stripped of a lot of removal, beast druid's regrowing boards place a great obstacle for such decks to overcome.

Noteworthy cards:

Bottomfeeder - This card is probably the most interesting new addition to beast druid and certainly impactful. With cards like Aquatic Form and Sir Finley, Sea Guide, you'll be able to refresh your board with growing aquatic beasts that quickly stabilize and maintain a huge board. Oracle of Elune again proves to be very valuable here making Bottomfeeder a good replacement for retired Umbral Owl.

Azsharan Gardens - I was surprised of how useful this card is in any game state. Especially with Aquatic Form, you'll quickly power up your minions' stats significantly. In my experience, just playing it for tempo even with only 1-2 minions in hand is good enough if Aquatic Form is available to follow up with a Sunken Gardens for a total of 2 mana.

Sir Finley, Sea Guide - If your hand really doesn't get you anywhere, which sometimes is just the case, Finley may turn an otherwise slow and lost game around providing you with fresh new cards as long as your hand wasn't empty.

Wildheart Guff - simply a great card. I am sure there are other versions of beast druid without him or Ivus, the Forest Lord, but in those match ups, this card has always proven helpful for its draw effect while ramp is always good to spend leftover mana.

Ivus, the Forest Lord - Well, Ivus, the Forest Lord + Oracle of Elune = great success when mana >= 10. In the end game, this can be very solid winning combo, especially with much of the removal gone atm. Ivus, the Forest Lord also offers a great swing in the midgame as it usually ends up a 5/5 - 7/7 with rush and divine shield to get rid of annoying obstacles.

Mulligan in about that order (bare in mind that it's too early to precisely predict the perfect mulligan):

Bottomfeeder
Jerry Rig Carpenter
Vibrant Squirrel
Druid of the Reef

Let me know what you think in the comments :)