I'm not sure I follow you.Anna was not married, and therefore not a Middleton when she had Arthur, if you believe what the census is telling us.The other person who replied to your message says he has the family line and Anna's maiden name was Martin.So therefore, Arthur is likely named Martin because he took his mother's name. (is that what you were trying to say?)
Here is the record of Anna and William Middleton's marriage, from Indiana Marriages 1845-1920.
County: Gibson Name: Anna F Martin Spouse: William H Middleton Marriage Date: 30 Jul 1906 Book: 11 Original Source Page: 343
Here are some census records I came up with.See what you think.The 1930 is the most interesting...
1920/IN/Vanderburgh/Evansville(PigeonTwp/ED118/p.58a Martin, Arthur/boarder/17/single/IN IN IN/waiter, restaurant
Evansville is not very far from Gibson County - we drive there from my husband's family farm in about 40 minutes.There was one other Arthur Martin, age 16, in the index but he was the nephew of a Arch Blair, living in Hartsville, Bartholomew County, IN and I don't even know where that is.I also found Anna:
1920/IN/Gibson/WhiteRiver/ED28/p.12A Middleton, William H/head/62/married/IL NJ IL/drayman, city hauling Middleton, Anna S/wife/38/married/IN IN IN
I found the Arthur below in 1930 in CA - if it's your Arthur it appears he may have had another wife and family at some point.When did he marry Vera and when was your mother born?
1930/CA/SanDiego/SanDiego/ED129/p.6A taken April 10th Martin, Arthur E./head/27/married at 24/IN IN IN/sailor, US navy Martin, Madeline/25/marr@22/CA CA CA Martin, Elizabeth A/6mos/CA IN CA
I could not find another Arthur in the 1930 census, born in Indiana that was a close match.I thought the one above was particularly interesting since he was in the service, which you had mentioned. Nancy