Is there a connection between "pork barrel" and "gravy train"?

Solution 1:

Both phrases have come about independently and have different meanings.

Etymonline.com:

gravy
Meaning "money easily acquired" first attested 1910; gravy train (1927) was originally railroad slang for a short haul that paid well.

More history here.

thefreedictionary.com:

gravy train
n. Slang
An occupation or other source of income that requires little effort while yielding considerable profit.

Etymonline.com:

Pork barrel "state's financial resources" is 1909, on notion of food supply kept in a barrel (lit. sense from 1801); the shortened form pork in this sense is attested from 1862.

thefreedictionary.com:

pork barrel
n. Slang
A government project or appropriation that yields jobs or other benefits to a specific locale and patronage opportunities to its political representative.