It is my understanding that it is entirely permissible;
It is permissible, however, to invoke the Name of Hashem to praise or to give thanks to Him. Similarly, it is permissible to call on the Name of Hashem in the context of private supplication. The familiar יהי רצון מלפנך ה' אלקינו as a prelude to prayer is acceptable and does not constitute uttering the Name of Hashem in vain (Rabbi Forst, The Laws of B'rachos, p. 48, with the footnote "Tosfos Chaim on Chayei Adam 5:7 citing the Magen Avraham 188:11. See Sefer Chassidim cited in M.B. 5:3")
(I believe that the issue in the question referenced is not, strictly speaking, one of a beracha l'vatala (a blessing in vain) but rather a berachah sh'aina tzirchah (an unnecessary blessing, using G-d's name in a formal blessing which was redundant or not obligatory), see ibid, page 55 for a better presentation.)