I see you have the birth certificate for Cecil. He probably didn't move on his own at age 2, his parents apparently moved . I did not find them at all in the 1910 census. As far as court records, at age 0 to 10, he could not have petitioned the court on his own behalf to change his name, not until age 21 (then). If his parents had so changed it legally, the below will help. But, in 1908, it was very rare to have an official name change for a child, many birth certificates indicated "Baby" and the name decided upon later. This Cecil could have almost immediately been called Thomas by either the father or mother, or both, on behalf of a parent, sibling, or preference. No one would consider any problem with him BEING Thomas, except perhaps if he joined the military, etc. Then, all he would have had to do would be bring documentary proof, school records, baptism records, signed depositions, etc. that he was Cecil. No one cared back then. Today you have to prove who you are for almost everything.

To answer your question:
1. The Court Clerk for the Superior Court of Okanogan County would be the initial repository for information concerning name change approvals:
http://www.okanogancounty.org/Clerk/Records%20%26%20request....
But, expect to pay, IF they have the record immediately available. IF they have a microfilm of such documents, at this time, you're talking a time period of 10 years.

2. Washington State Archives, Central Regional Branch: Okanogan
http://209.85.173.104/search?q=cache:CJFtDlV5UFkJ:www.cwu.ed...

Since this link may not work. The Washington State Archives, Central WA Region, has archives relating to Okanogan Superior Court. I looked through the available file listings, but did not find any files relating to name changes. Another source would be the County Recorder...IF they recorded the name change. This current link will work if you want to navigate it to the Okanogan holdings. You can contact them for information. Should they identify a possible record repository, if you do not live near the facility, you might enlist a local genealogy researcher.

My search findings:

The Washington Birth Index shows Emma Beeman. and Ammond Beeman (Twins), born 9 July 1913 in Clallam County to E.G. Beeman and Cora Violla Wilson. So this shows the parents of Cecil apparently moved to Clallam County by 1913.

The World War I Draft Registration Card Image database shows Egbert Garrett Beeman, born December 13th 1876, living in carlsborg, Clallam Co, WA

The 1920 Census for Reno, Clallam County, WA, enumerated 9 January.
Beeman, E.G. age 43, born in Idaho, to father born in the US, mother in Missouri, teamster in a saw mill.
Cora, age 43, born in Iowa to father born in Indiana, mother in Missouri
Ethel, 13, born in WA.
Thomas, 11, born in WA
Amon, 6, born in WA (Twin, Emma, not shown)

Ron Bestrom