To this day, no letter between the Little family members has been found, either before or after Edward's disappearance in the arctic, but some archives and newspapers give up a glimpse of how the expedition affected the family life
The Relief Expedition meeting
In 1857, James Cornelius Little, Edward's eldest brother, took part in a meeting centered around Lady Jane Franklin's effort to send another expedition in search of her husband and his crew. James Cornelius was described as "an esteemed resident" [of Jersey] and the he "had taken a lively interest in the proceedings of this meeting. He had to deplore the loss of a brother in the ill-fated expedition of Sir John Franklin. He was happy to have the opportunity to express his thanks to the gentlemen who had taken the initiative in the proceedings." James Cornelius also subscribed to the cause by donating 10£ ¹

Jersey Independent and Daily Telegraph, 27 June 1857
The Arctic Medal
In 1857, the first polar award was instituted : the Arctic Medal. It was "Awarded
retrospectively to all officers and men engaged in expeditions to the
polar regions from 1818 to 1855. It was granted to members of the Royal
Navy, civilians, scientists, personnel of the French and US navies and
employees of the Hudson Bay Company who took part in both exploration
and rescue parties, including the abortive searches for Sir John
Franklin and his crew who were lost on the 1845-8 expedition. 1486
medals were awarded of which 1106 were received by officers and men of
the Royal Navy." ²
In 1904, the Polar Medal was instituted
From the archives, we can see that Edward was awarded one ³, it was sent to his sister (it does nos specify which one, but it could have been Elizabeth Jane, the eldest)
"Sent to sister 23/7/57"

ADM 171/9, Arctic Medal Roll of Recipients, 1818 - 1855, The National Archives
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First edition of the Polar Medal, "the Arctic Medal" ⁴
Edward Little's Epitaph
Edward's parents, Simon & Sarah were buried in Green Street Cemetery (Section Roseville Street, Grave YY2, St Helier, Jersey) with their eldest daughter Elizabeth Jane. Following their own epitaph, the family added a few words to commemorate Edward ⁵
It's the tombstone on the right, you can see the names if you zoom in,

Green Street Cemetery, St Hélier, November 2008, Wikimedia Commons
It reads :
"To the memory of
Sarah,
the beloved wife of
Simon Little
Paymaster RN
who died 9th May 1847
aged 68 years
Also of
Simon Little
husband of the above
who died 20th July 1852
aged 84 years
And of
Elizabeth Jane Little
eldest daughter of the above
who died 27th May 1874
aged 70 years
Also of
Edward Little
their son, Commander RN who was lost in
the Arctic Expedition under
Captain John Franklin RN
[unintelligible] HM Ship Erebus 1847
The Greenwich Memorial
The chapel memorial, Greenwich, 2024, photo taken by the author
Right at the entrance of the Chapel of St Peter and St Paul of Greenwich, you can find a marble memorial commemorating the Franklin Expedition. Installed in 1869, (it was recently moved to its current position), the monument holds the remains of one of the men. Repatriated in 1873 and initially thought to be the remains of Erebus' second lieutenant, Henry Le Vesconte, a recent study claims these are more likely the remains of Assistant Surgeon and Naturalist, Harry DS Goodsir. ⁶
"As you look at the memorial, the two sides symbolize hope at the outset of the expedition followed by despair as
the expedition meets its tragic end. On the left a naval officer studies an open book, with compasses in hand. Near him are a globe and papers relating
to Arctic research, inscribed with the names of Franklin, Ross and
Parry. In the background are the tall masts of a ship, with sails set as
if about to depart. To the right, in
the sky above the towering, jagged icebergs rests the North or Polar
Star. Below, a desolate sailor sits on a fragment of rock. One of his
feet is bandaged and broken equipment litters the scene. "
Edward's name was inscribed as Commander Edward Little, he was promoted in absentia in November 1846 while on the Expedition. He likely never knew about his promotion, as the last correspondence with HMS Terror was in 1845
Simon Little's letter
The UK Hydrographic Office (UKFO) holds several letters from the families of the lost men, inquiring about the whereabouts of the rescue missions. One of which was written by Simon Little, Edward's father, in 1851, requesting a polar chart of the proceedings of one of said rescue missions ⁸
The letter was fully transcribed by Edmund Wuyts on Arctonauts ⁹
It's also worth mentioning that even though at least three of his brothers were
still in the service of the Royal Navy or the Royal Marines at the time,
I couldn't find any mention of them taking part in any rescue
parties in search for the lost men of the Franklin Expedition ¹⁰
Sources
¹ Jersey Independent and Daily Telegraph, 27 June 1857
³ ADM 171/9, Arctic Medal Roll of Recipients, 1818 - 1855, The National Archives
⁴This article uses material from the “Polar Medal” article on the Franklin Expedition wiki at Fandom and is licensed under the Creative Commons Attribution-Share Alike License.
⁸ LP1857/L/22, Incoming Letters prior to 1857, Letter from : Little, 1851, UKHO
¹⁰ ADM 171/10, List of officers and men of the Royal Navy and Royal Marines employed on Arctic, 1818 - 1855, The National Archives