Three people were on board the Merlin Mk4 when it ditched off the Dorsetcoast.
The other crew were rescued and taken to hospital, but the Ministry of Defence said their injuries are not serious.
“Rhod was so immeasurably loved by his parents, siblings, partner, friends and family and he was devoted to them. Our lives will never be the same without him,” said a family statement.
“We are all so very proud of the talented, passionate, strong and loyal man he was. We will always have him in our hearts. Our wonderful boy.”
Lieutenant Leyshon was a “shining light with a bright future”, said commanding officer Colonel Mark Johnson, from Yeovilton airbase in Somerset.
“He epitomised our team spirit and his loss leaves a huge hole in all our hearts,” he added.
Lt Leyshon joined the Wales URNU (University Royal Naval Unit) in 2010 and was commissioned in 2014.
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The Royal Navy described him as one of its “most trusted and highly capable aircraft captains and instructors”.
He served with 845 Naval Air Squadron and had been deployed to the US, Caribbean and Norway.
For the last 18 months, he served with 846 Naval Air Squadron.
“I flew with Lt Leyshon just a few days ago and personally attest to both his professionalism as a pilot and his warmth as a character,” said Air Vice-Marshal Alastair Smith.
“He will be sorely missed by the squadron, the Commando Helicopter Force and the wider Joint Aviation Command.”
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Last movements of helicopter before crash
Tracking data showed two Merlins from Yeovilton operating off Dorset on Wednesday, with both airborne at 8.40pm but disappearing by 8.52pm.