It's hard to prove an absence of evidence, but there are good reasons why George I might notnot have considered changing his name: it was a common given name in George's family, and then there is the obvious link to St. GeorgeSt. George, patron saint of England. Further, there was no precedence (in England at least) at the time for a monarch changing name, and George was already a popular name in England.
Perhaps a further point to consider is that there appears to be no precedent for an English or British monarch to change his or her name upon ascending the throne. You cite the Scottish example of Robert III but, as you pointed out, he had good reason not to use his birth name. Note also that the English monarchs who did not use their first given name as their regnal name (Alexandrina VictoriaVictoria, Albert Edward VII**Edward** VII and Albert Frederick Arthur George VI**George** VI) did use another of their given names. George I's only other given name was Louis or Ludwig; why use one of these when George was already a popular name in England?
As for options, you mention James. However, this would most likely have been considered inappropriate given the divisions caused by James II'sJames II's Catholicism and the emphasis on a Protestant succession to AnneAnne. Henry would have been better, but Henry VIII'sHenry VIII's six wives would not have been seen as ideal (and all the Henrys before him were Catholics). More promising would be Edward, given Edward VI'sEdward VI's impeccable Protestant credentials, or perhaps William, but I can find no evidence that these were considered.