@Brandon Turner I've not been listening to the podcast/been a part of the site long enough to have heard about the Kurt Cobain house, but it sounds like there's an awesome story there. As for your tenant, you have a few options. Whether they work depends entirely on how your local PHA interprets the rules and views the needs of your area.
#1 Ask for a "Reasonable Accommodation". Perhaps this little old lady is disabled, perhaps she's next door to family who helps caregive for her, perhaps she's mobility impaired and is next door to her doctor, or perhaps there's another aspect to the unit that makes it uniquely appropriate for her that wouldn't be easily replicable elsewhere, or at the traditional market rent. You can speak to your PHA about their Reasonable Accommodation process and find out how to make your case in writing. Asking for 10% or less above FMR is the easiest for them to approve, more than that requires extra work/approvals on the PHA end, thus are harder to win.
#2 Play with the Happy Assistance Estimator (which I believe is in the downloads section of BP) with the FMR and values for local utility allowances to see if including some or all utilities will up your total cashflow. Since she's an older lady, in my experience single old folks tend to sip utilities, rather than hog them, so there might be a margin you can gain there. Its a generalization on usage, YMMV, but its an option. If you'd like help figuring out how to use the estimator, PM me. Its not incredibly intuitive.
#3 A borderline legal option is for when you redo the lease, take out an amenity such as use of the laundry room or parking space, and charge separately outside of the lease agreement for that service. Federal law prohibits charging extra for rent/utilities outside of the lease agreement approved by the PHA, but there is grey area about extras like laundry or a parking space if its outside of the unit. A word of caution though, the PHA will likely take a dim view of such an arrangement if they find out, so make sure everyone is on the same page.
#4 File an appeal. PHAs have some flexibility with FMRs to ensure tenants are able to stay mixed into broad sections of their PHA area, rather than just congregate in low income neighborhoods. Not all PHAs have an appeal system to work on this issue, as its not an issue in all areas (for instance, more rural PHAs tend to have a much more homogeneous rents across their coverage area), but it could be an option in your area. Again, the 10% and under = easier approval rule usually applies here as well.
#5 Take the lower rent. She sounds about as rock solid as a tenant can be, so giving up 10% each month instead of perhaps an average of 10% over the next ten years in vacancies/unpaid rent would be fine in my book.
#6 Raise the rent after 1 year. Most PHAs won't require someone to leave their unit even if a rent increase pushes them over the FMR for their area, they will simply offer the option to the tenant to pay the difference or move to a more affordable unit. This is another tactic that PHAs take a dim view of when done repeatedly, and has encouraged a few to institute rules/policies where they will not continue to allow someone to use their voucher in such a circumstance unless approved. This is something you should ask another local landlord that's tried it about, as asking the PHA how to best circumvent their rules is probably not the best option, haha.
Hope this helps, good luck!
(edited for typos)