That depends on what stage you were at. If you both had an idea to 'do search' and had only got as far as picking a name. Then after another 10years of work by you alone you are ready to take on Google, does your late partner's partner get half?

If you need to bring in investment or new people and give them options who has to agree to that dilution?

If in a few years there is a row in the family about who owns your late partner's share - are you ready to spend time and money to defend in court what you said to who at the funeral?

Again my condolences for the OP's loss. BUT Startup founders need to consider that today they are co-owners with their best friend - tomorrow they may be working with their friend's newly divorced and bitter wife, their useless kids or a slimey lawyer who is preying on their grieving parents.