You would have to formally amend the trust. This can be done by a revocation and amendment that would be one page whereby each settlor declares the original trust revoked, and then amended to add the new successor trustee's name, and then states that with the exception of the addition, the original trust remains in full force and effect.
And, then you would both sign before a notary public.
A handwritten amendment would likely be interpreted as invalid, because for starters, only one settlor's handwriting would be present -- which would only show one settlor's intent.
Of course you could both write the exact same thing onto the face of the trust, but it could be pretty difficult to squeeze things in, effectively.
Also, to be frank, son-in-laws sometimes become "ex" son-in-laws. So, while I'm sure you have nothing but the best intentions here, be careful about naming an "in-law" as a trustee, because I have seen this sort of thing go awry before.
You could probably get a lawyer to write a one-page amendment for maybe $150 -- if your only change is to add the successor trustee and delete the deceased child (my condolences, by the way).
For an estate planning
attorney referral, see this link.
Hope this helps.