As Jim said, there are a lot of posts on 'drywells' and 'french drains' on the Forum. Use your FIND / SEARCH feature in your Browser to search the page for those key words (or search for 'partial' keywords.)
Yes, it's possible that a drywell is all you need. And yes, it's size depends on how much water you get off your roof. Measure up the house, and 'note' which roofs are feeding the 'gutters and downspouts' that are affecting your neighbor. (It's possible that only the BACK part of the roof, for example, is giving your neighbor the problem; NOT the front roof. See what I'm saying?)
Then, go talk to to someone at your local Quarry or Gravel & Stone Yard. (They're listed in the Yellow Pages under QUARRY and STONE CRUSHED and SAND & STONE. (The Yellow Pages has an index for more help.) They'll be able to give you an idea of how much 3/4" crushed stone (gravel) you need and probably how big of a well you need.
One last, important 'thing'. Be sure to line your well (and/or your French Drain) with Filter Fabric. That way, the dirt that surrounds the well doesn't contaminate your gravel. Otherwise, you'll have that AND settling problems later on. Also, remember, you needn't try to be able to hold ALL the water that comes off the 'affected' part of the roof to help your neighbor. 1) Hopefully, the ground will help absorbe the water as it runs into it, and 2) hopefully, your neighbor's lot can handle SOME of the water when the well 'over-flows'.
It's nice you are wanting to help him out. Be sure to get him to kick in with some labor or some $$$. My best to ya and hope this helps.