My ex-wife is no longer on my checking account, but I have still been using the checks with no problem. Her name is written below mine. I want to know if I take a marker and black out her name will the bank think something is fishy and deny it? Doesn't really seem like an important part of the check. Yes, I know I could just keep doing what works until the checks are used up, but in this case I would like to marker her name out.
If it were me, I would get a new checking account at potentially a new bank, but certainly with a new account number.
As Nathan said, there is no need for you to cross her name off the check, but potentially, she could use those checks, or have new checks printed to use. Having her name on the check makes it seem like she is a legitimate signer on the account.
In the end you can fight and possibly win with your bank that they should not have accepted a check signed by her as payment, but why bother? Also you will potentially alienate any merchant that accepts a check by her. It is a total mess that can be relatively easily solved with very little money ($25-$40 for check reprinting) proactively.
Close the account, shred any existing checks, and move on. Heck you can actually make money by doing this and receiving a bonus. Check Nerd Wallet for current bank promotions.
If it doesn't seem that important, why bother blacking the name out? For the effort, it might cost you less in your time to have the checks reprinted.
There's no way to know what all banks would do with a check that has a name crossed out, but most would ignore it. Most checks are processed automatically. Signatures are not verified, post-dated checks can usually still be deposited. Occasionally you'll have a bank or merchant reject a check, but don't expect that to be the norm.
For what it's worth, 20 years ago I had a huge box of sheet-feed checks (3 or 4 per sheet) - After the divorce I kept the account, then ran the remaining checks through a Word doc that tidily blacked out the ex's name. Never had a hitch, glitch, or rejected check.