Short answer
It's hard to prove an absence of evidence, but there are good reasons why George I might not have considered changing his name: it was a common given name in George's family, and then there is the obvious link to St. 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.
Details
The name George (as noted by Generalissimo in his comment above) was a popular name for the House of Welf (or Guelf or Guelph). This, along with the link to St. George, is noted by Tim Blanning in his chapter in The Hanoverian Succession:
It was also a happy accident that the favourite first name of the
Guelphs was ‘George’, allowing the pamphleteers and cartoonists to
cross-refer to St George slaying the Catholic dragon. The author of
‘The Welcome’ (to George I), for example, revelled in the terror
struck into the hearts of papists by the realisation that:
Their Popish Dragon now must lose his Sting,
Because St George our Champion is, and King.
Source: Tim Blanning, 'The Hanoverian Monarchy and the Culture of Representation'. In Andreas Gestrich and Michael Schaich (eds.), 'The Hanoverian Succession: Dynastic Politics and Monarchical Culture' (2015)
Perhaps a further point to consider is that there appears to be no precedent for an English 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 Victoria, Albert Edward VII and Albert Frederick Arthur 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?
There is a commonly held view that George was little used in Britain
until George I ascended the throne in the 1714. This, however, is
simply not the case. In fact, George ranked within the top 10 most
popular boys' names in England from the early 1500s at least, as we
can see from Smith-Bannister's rankings
According to these rankings of most popular boys' names, George ranked between 6th and 10th for every decade between 1538 and 1700, and it's popularity continued until the 1930s.
As for options, you mention James. However, this would have been inappropriate given the divisions caused by James II's Catholicism and the emphasis on a Protestant succession to Anne. Henry would have been better, but Henry 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's impeccable Protestant credentials, or perhaps William, but I can find no evidence that these were considered.
Whatever the name, though, I can find no evidence that George (or anyone else) ever considered anything other than George. There is also no mention of George considering a name change in Ragnhild Hatton's comprehensive Yale English Monarchs title George I (of which I have a copy).
Other source:
Hannah Smith, Hannah Richardson, 'Georgian Monarchy: Politics and Culture, 1714-1760' (2006)
