I think since she wanted to move house and he didn't, she wrote "Move On" or something along those lines which made him realise he was stuck and finally let go

In an interview with the director, he explained that it didn't really matter to the story. Rooney Mara did write something on the paper, but no one ever knew what it was. Earlier in the movie her character said she liked to leave notes to remind herself what she liked about the house or something like that... I think that her character would have written something about it being the home she shared with her husband and that he was loved & missed.

I like the interpretations above, but I found myself not really caring at all about what was written... It seemed both inconsequential to the themes of the movie, but more troubling, I found myself indifferent to this and wouldn't have cared even if it dod matter...

I think that was a sign that the movie didn't work for me... It didn't resonate...

I assumed the note said "love", but I'm sure it's projecting. In fact, she said something like "little rhymes or poems...remembering living in the house or what I liked about it."

One key though is when he asked her if she ever went back to read them she said," No, I didn't need to." Perhaps this compels his motivation? I get why leaving the note mysterious was the right call, but can't say I wasn't disappointed. I presumed we'd at least get a partial read as it lay on the sheet.

Also interesting to note that the two times the house "opened up" was right as he dug the note out of the wall...the second being an open door to a beautiful day.