My mother hasn't given me much information, she's as surprised as anyone how much I'm finding on the intranet.
A lot of my information is using the names I know along with a period of time and just trying to get a hit. For example:
I did a search on "Franklin Lawrence", not "Frank", figuring in his youth his parents might have used the longer form of his name. I came across a Franklin Lawrence (age 12) living in the Cady township in St. Croix county in 1880. That's the same township Avery Lawrence and family were living in around 1905. I'm assuming this is the "Frank" I'm looking for, just due to location and proximity of a future event. This "Franklin Lawrence" also had two younger siblings at that time - a sister named Laura and a brother named Charley.
Further research of this individual by the name "Franklin Lawrence" shows he was living in Iowa in 1870 (at an approximate age of 2 - another strong indicator as Frank D. Lawrence was born around 1869) with parents by the name Jasper and Almira. These two eventually settled in the Spring Valley area and were last living there in the 1910 census. Coincidently, Spring Valley is where Avery was born. Again, all guesses on my part, but the puzzle seems to be coming together.
The Chase side I did the same, and came up with May's parents - Ira B. and Kate E. (I have no idea what her maiden name was). Kate was from England, just like May indicated in many of her Census forms throughout the years - again I'm guessing. I inquired of these two individuals with some churches in Ladysmith, but have heard nothing back.
Futher back than that I haven't tried to go, as the trail gets a lot colder.
Good luck with your search. I know guessing is never the way to go, but sometimes you need a little faith to get over the next hurdle.