Get an equivalent to what he played and work from there...  It looks like he played the range of banjos; it's how, not what.

Dock Boggs (1898–1971) famously played a Gibson Mastertone banjo, which he purchased after his initial 1920s recording sessions, often citing it as "the best banjo he could buy". Earlier in his career, he played a Sears & Roebuck banjo. He is known for a unique, haunting style that combined Appalachian folk with African-American blues, utilizing heavy-gauge strings. 

Key details regarding Doc Boggs' instruments:

• Gibson Mastertone: Purchased following his 1927–1929 Brunswick recording sessions, this was his preferred instrument for much of his recorded legacy.
• Sears & Roebuck Banjo: Used during his earliest, formative years and in the early 1920s.
• Strings and Setup: According to folklore, he used very heavy strings, often referred to as "Gibson medium" (e.g., .012 to .025 gauge), to achieve his distinct, dark, dissonant sound.

A used 70's-80's Asian-made "Masterclone" ought to turn up now and then at that price, if you keep an eye out.

Dock liked those heavy strings with a wound 3rd, but he also tuned down a whole step or two below standard, as I recall.

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Originally posted by Dan Gellert

Dock liked those heavy strings with a wound 3rd, but he also tuned down a whole step or two below standard, as I recall.

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Probably a calf-skin head too; suede is a fair synthetic option.

Second what Dan said, Japanese Masterclone, look for one with an archtop tone ring,

For comparative purposes, "Davenport" is played on a 1974 Taka RB250 clone that I fitted with a Stewmac 20 hole flathead tonering, you can buy these for $400 or less as Dan pointed out. "Country Blues" is played on my go-to 1974 OME "X" with a 20 hole flat top tonering of unknown origin that I got off of ebay for $700 and then restored. I use medium goatskin heads, and as Dan said a wound .018 third string (in addition to heavier strings in general).

I don't know what they go for anymore, but an RK Madison with the wooden tonering might work for you too. Dock played resonated banjos after 1927, but the original Brunswick recordings were played on an $18 (1920 pricing, about $300 adjusted for inflation) Sears 'Supertone" open back. After his sessions, he took the money and bought a Gibson Mastertone.

Happy hunting.