Assuming the photo is in the 2:3 aspect ratio (same shape as 35mm film and many digital sensors), then I suggest a resolution of 1200 x 1800 pixels. That is enough pixels to show at a decent size on most screens, and also allows one to make a nice 4" x 6" print and a little bigger in many cases.
If the image is exported as a JPEG, Quality = 85, the file size will be something around 0.80 to 1.6 MB which should not be a problem for transferring by internet.
RE: "... what is the largest image size that will not overwhelm." - three possible bottlenecks to consider:
Your internet service provider or email service may have a maximum file size that will block emails if they include an image that is too big.
Her internet service provider or email service may have a maximum file size that will block emails if they include an image that is too big.
If either party does not have broadband, sending multiple large files may be allowed, but may slow down the user's computer/phone etc. while the images are being sent or received.
I have not investigated the finer points of sending photos by text (multimedia messaging), so there may be another set of rules to consider for texting.