First of all.. I am sorry to hear about the loss of your father.
In respect to his Smith, it is indeed an early one and looks to be in very good shape. Many Smiths are found with some or substantial amounts of finish on them, but there are certainly many well used examples as well. This one doesn't appear to have any finish remaining, but wood looks great and cartouche (indicating acceptance by US military) is crisp. Being an early example by serial number, this would have been delivered in early/mid 1862 from my understanding. I am fond of the Smith but could not call myself an expert by any stretch of the imagination. I am primarily going by info found in the book, "U.S. Civil War Carbines" by Whisker, Yantz, and Hartzler.
As far as value, I would say retail would be $1300-1500 range but that is just a guess. Even though the Smith was highly thought of during the war it doesn't seem to be a top collectible by CW firearms collectors. Some may disagree, but I am just going by how long they seem to sit around in dealer website/inventories compared to Sharps or Spencers. I mean no disrespect, and it looks to be a great example that anyone who collects CW firearms would be happy to own.
One additional note,... in searching know serial numbers to units, there is some data on the Smith but from my records this one isn't a direct hit. There is one (serial number 444) that went to the 1st Conn Cavalry. The entry is just dated 1863 but it doesn't necessarily mean it was issued then. Even though the serial number is close it doesn't mean it went to same unit. Many make this mistake of "range matches" but imo that is misleading at best. It isn't as though they took serial numbers in perfect sequence and sent them to the same units.
Also, there was a recent discussion about the Smith here that you might find interesting...
At 30,000 the Buttermilk Falls Blaster was the fourth most used Union carbine. Some troops liked the Smith Carbine and found its durability and ease of use, while other troops did not care for it. So was the 'Buttermilk Falls Blaster' a good carbine?
civilwartalk.com
Regards.