As the answer from @StephenTG states, the secret is to
interpret the grids as cells in Conway's Game of Life (a thought I'd had, and intended to investigate further this evening)
Specifically,
it is run on a finite 5x5 grid where all cells outside the 5x5 area are considered to be permanently 'dead' (one common alternative is to run it on a toroidally-connected grid, but this is ruled out because several of the patterns shown would have different behaviour on such a grid).
Implementing the necessary calculations in Excel:

We can see that, as also stated in @StephenTG's answer,
Taking $N$ as the generation where a stable configuration is reached, and $K$ as the number of live cells in that stable configuration, the final answer adds $N + K$.
For starting grids that reach no stable configuration, $N = \infty$
Higher finite scores are possible. For example,
I was able to quickly construct grids which score $13 + 4 = 17$ and $3 + 16 = 19$


... and revisiting it a little later, some minor tweaks improve this:
$27 + 6 = 33$

Later, I finally got round to doing an exhaustive computer search for better solutions. The most relevant part of the output
shows both the longest-lived starting state, and also the highest-scoring (subsequent generations are left as an exercise for the reader):
State 257296 : 39 + 0 = 39
[]
[]
[][] [][]
[][][]
New best score: 39 + 0 = 39
State 12366675 : 34 + 6 = 40
[][] []
[] [][]
[][]
[] [][]
[][] []
New best score: 34 + 6 = 40
Search Time: 35.3581088 seconds
Showing 48 states with best score (40):