Great - it looks like your post generated some new information and your family information is beginning to come together!I used familysearch.org and census records at ancestry.com for the information.One thing to keep in mind when using census records is that names and ages were not always recorded correctly and sometimes you have to get creative when searching.I was able to find more on Winnie's line.The Hollerans can be found in the census records for Castleton 1850-1880 (when Frank Briggs was recorded living with grandmother Bridget Holleran).The surname is found in census records as Halran, Holland, Hallman, any variation you can think of.I was able to track them by searching just for Bridget, b + or - 2yrs 1820, Ireland, living in Castleton, VT.In 1850, I found Matthew Hallman, age 36, b Ire., wife Bridget, 30, b Ire James, 14 Judy, 11 Bridget, 15 Winnifred, 7 Mary, 5 Michael, 3 Catherine, 1 Matthew, 0 (newborn)
In 1960: Bridget Hollan, 40, b Ire. head of household Mary 14 Michael, 12 Catherine, 11 John, 9 (likely the John Halran rec. with Winnie Briggs in Ohio in 1870) Jennie, 7
Using the same sources (familysearch.org and ancestry) I found that Lyman Briggs was b 5 Jun 1804 Goshen VT; bp 10 Dec 1858 at Brandon VT; mar. 25 Mar 1825 at Brandon Sibel Stanley Hale;Lyman was the son of William Briggs and Chloe Tuttle.
I have a lot of Briggs resources so I will see if I can find any further information on this Briggs line.