I'm guessing the East is because it appears he may have been the illegitimate child of Mary East before she married John Eltham (see one of LizzieL's posts above)
Looks like Henry may have had a little bother - some snippets from Biritish Newspapers Online freesearch. I've no sub, so not sure if anyone else can check them out as they might be useful
"Before Judge Parry.— Henry Eltham, travelling confectioner, ... ”
Saturday 11 October 1873 , Northampton Mercury
also one where he's the victim
“... October last, feloniously entered the dwelling-house of Henry Eltham. at Ensham, and stolen thereout 7 sovereigns, 17 half-sovereigns, crowns, and other articles, ... ”
Saturday 17 July 1841 , Berkshire Chronicle
and some for Emily
ABINGDON
“... than 20 years. As there appeared some doubt in the matter, the bench remanded the prisoner, and bound him over to appear. Emily Eltham, the wife of a confectioner ... ”
Saturday 15 May 1852 , Berkshire Chronicle ,
"CITY COURT.-TuESDAY. (Before the Mayor Ald. Thorp, Ald. Towle, and Mr. Justice Pike.) Emily Eltham, the wife of Henry Eltham, of Eynsham, was charged ... ”
Saturday 22 May 1858 , Oxford Journal
and again as a victim (possibly)
Reading Mercury
WESTERN CIRCUIT—Winchester, July 11
“... court. {Before Cfiief Justice Tindal.) Joseph Williams, aged 26, was indicted upon charge of having feloniously wounded Emily Eltham, with intent to maim her. The ... ”
Saturday 20 July 1844 , Reading Mercury