This answer is based off of experience of being present at a giyur l'chumra conducted by R' Yosef Berger. The Beis Din does not remind the convert, as is normally done, that this is his last chance to change his mind and not become a Jew, as we are already assuming that he is a Jew, and the conversion is "just in case." In the conversion which I witnessed, they did not inform him about mitzvos kalos v'chamuros (some "light" and some "serious" mitzvos), for what I assume was the same reason - because we don't want to and aren't trying to scare him/her off.
Normally, a convert makes a blessing on going to mikvah, and a shehecheyanu. The giyur l'chumra convert makes neither of these blessings, although I suppose it couldn't hurt for him to bring a new fruit with him and just have it in mind.
The convert was not asked to choose a name for himself, as normally converts do take a name upon conversion. I do not know if this was because it is assumed he wouldn't want to change the name he had been going by as a Jew, or because we aren't interested in changing his name since we are assuming he already was a Jew.