Can I have Hashem’s name as the background of my phone to influence me to be a better jew and to always have Hashem’s name in my head?
ב"ה Shalom Your desire is to be highly respected and commended since it is important to have Hashem's presence felt throughout our day and Hashem's name spoken on our lips. (See: מגן אבות לרשב"ץ על אבות הקדמה, של"ה שער האותיות אות אל"ף - אמת ואמונה ד"ה מט) However, due to the reverence we are required to give Hashem's name, my recommendation would be not to put Hashem's name on your phone, because this may cause disrespect to Hashem's name. This is the ruling of the Rema in Yoreh De'ah 276 and others.יור"ד, סי' רע"ו: יג, שו"ת הרדב"ז חלק א' סימן ר"כ, שו"ת יחוה דעת חלק ג סימן עח ) The simplest example is that we carry our phones wherever we go and in some places, it is simply forbidden and inappropriate that Hashem's name appears. I may suggest, that you place the picture of a great Rabbi who you respect and look up to. That can also instill a feeling of Yirat Shamayim= fear of Hashem. Needless to say, that to instill Yirat Shamayim, a book like Mesilat Yesharim of the Ramchal is very important. All the best