Q. Would one be allowed to compose their own teffila, and if so can one say Hashem’s name in it?

A. Midrash Tanchuma (Bereshis – Vayeshev 8) comments on the verse (39: 3) “And his master saw that the Hashem was with him,” how would the Egyptian master know? And explains that Yosef would constantly pray to Hashem and he overheard him. So it would seem from all the tefilos quoted in Tanach by the Patrirachs, Moshe Rabbenu, Eliahu, Chana,the Prophets et. al. And the tefilos of tzadikim throughout all generations are legendary.
Horav Shlomo Miller’s Shlit’a opinion is similar and one should always pray to Hashem when in need or to express thanks, even using His name with the honor due.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a