Nikon 55-300mm or Nikon 70-300mm which lens is a better choice as a present for a beginer photographer?

Realize that there are two versions of the Nikon 70-300mm lens. The $172 Nikon "AF 70-300mm f/4~5.6" will not autofocus on your brother's D5100, but the $589 Nikon "AF-S Nikon 70-300mm f/4.5~5.6" will.

The $399 Nikon "AF-S 55-300mm f/4.5~5.6" will also autofocus on your brother's D5100.

Basically, the D5100 requires "AF-S" lenses to autofocus. The $172 version of the 70-300 is an AF, not AF-S version.

The primary difference between the AF-S 55-300mm and AF-S 70-300mm are not significant for your brother's camera. However, if your brother ever decides to upgrade to a full frame DSLR, the 70-300 would be the better one as it is a full frame (and larger/heavier) lens than the 55-300 (which is a DX lens).

The 55-300 would probably be the better choice for your brother's D5100.

For the Nikon D5100, and all other Nikon entry-level DSLR's, you'll need a lens that's AF-S, signifying a built-in autofocus motor. The new Nikon 55-300mm AF-S VR has that, as does the Nikon 70-300mm AF-S VR. Be aware that Nikon also makes a 70-300mm AF-D and a 70-300mm AF-G that won't autofocus with his camera body.

Between the two lenses you named, each has advantages.

The 55-300mm is newer, lighter, smaller, less expensive and has a wider zoom range.

http://bythom.com/nikkor-55-300mm-DX-lens-review.h...

But the 70-300mm is a full-framed lense, able to work with any FX Nikons in your brother's future. It also has a faster zoom motor, important if he's going to be using this for sports.

http://bythom.com/70300VRlens.htm

Either lens will be a great addition to his camera bag, and he's fortunate to have a brother like you who's so considerate and generous.

Nikon 70-300mm AF-S. The shorter the zoom, the better optical quality is and the AF-S spec will make sure the lens will autofocus with the D5100.

The 55-200 is probably a better (and cheaper) choice than either.

Whichever lens you get, make sure it's AF-S - the older AF type require a body motor for autofocus, so AF won't work on a D5100 which relies on a lens motor.