Dealing with professor's death on my thesis

You should definitely keep the name. I've seen that people who passed away during some endeavor usually have their names surrounded with a rectangle.

You can also consider putting some In Memoriam page for your mentor somewhere at the beginning of the thesis. You should consult other people within your institutions culture to make sure that this doesn't come off as pretentious (although I see no reason why it should).

My sincere condolences. I am in the same situation: one of my advisors passed away 6 months before the end of my PhD.

Here are the issues I faced:

• My thesis had to be supervised until the viva. OP (as it was the case for me) seems to have another supervisor. Talk with them and this shouldn't be an issue — especially if this other supervisor is in the academia.
  The PI of my research group also offered me to review my thesis even if it was not really within its field of expertize. You might otherwise find a more adequate other researcher ready to help you in your institution (just make sure that everyone in charge of your graduating program is in the loop when contacting new temporary advisors).

• I had to put names on my thesis' cover. I just listed names of my two original supervisors. (I don't think that temporary supervising a thesis for two weeks is worth being listed as supervisor.) You can find here ideas on how to indicate your supervisor passed away.
  The solution I chose was to indicate nothing on the front cover, to add a dedication page ("In the memory of Professor John Doe") at the beginning of my thesis, and to thank the temporary advisor in the Acknowledgement section. In this section, I also refer to my deceased supervisor as "the late Prof. Doe".
  Reason for no special symbol on the cover is that your deceased supervisor was de facto your supervisor for your thesis. It thus must be listed. But my opinion is that you won't change your thesis cover in the future if another person passes away.
  So I think that the purpose of the cover is to indicate who supervised you (here, both supervisors, hence both names listed), not their current state (hence, no indication). The dedication page, imho, already makes obvious that this person passed away short before graduation.

• This supervisor co-authored papers that were still under review. With the consent of my second advisor, I sent a note to journal's editors at the next round of review to explain them the situation. I asked them if it was possible to keep him/her acknowledged as author. It is then up to editors.
  Some journals have strict policies (e.g., having the written co-authoring consent from all authors regarding the accepted version — which we obviously cannot have.) Their name was then listed in the acknowledgment section.
  Most of them don't. What happened for a conference paper is that the name was listed along with the others, with a footnote explaining that the paper was finalized after their death.

TL;DR: You should list both names in your thesis. Regarding indicating that one of them passed away, I would keep it short and subtle. (My opinion is to add no specific indication on the front cover — see above.) For what regards on-going publications, talk with the editors the next time you contact them.

In my country we put a frame, with the name and surname inside the rectangle. This means that the author died in the process of creation.