Seed beads are close to my heart, they're one of the first crafty things I ever got into. They're incredibly versatile, and in this case pretty festive, too. They work well with math, a lot of common beadwork stitches are influenced by fractal-like increases and, in this case, I'm working with the relationship between hexagons and circles. (Is there a special term for that relationship mathies? I couldn't find anything easily with google!)
Don't let the tiny beads intimidate you. Start with small eggs and work up to larger projects. I make up my patterns as I go, but you could also make up a chart and plan that in advance.
Step 1: Supplies and Equipment
Eggs - wood are nice because they're sturdy. Base coat them to match your beads. Plastic would work, real are kind of risky (though pysanky people use real eggs, so it works for them!)
Seed Beads - these are size 11, size 15 is smaller and therefore would be better but more tedious.
Beading Thread - this is nymo in size OO. Use a beading thread, not a sewing thread.
Beading Needles - I use size 12 needles.
Scissors - tiny points are helpful for trimming thread.
Step 2: Peyote Stitch
One of the classic beading stitches is called peyote. It works up quickly (for beadwork.) You add a bead to every other column in each row, and the beads end up in a brick style layout. It is well suited for diagonal and vertical lines, horizontal lines not so much. If you've never used this stitch before google it and make a few things with a flat, even number of beads peyote.
Step 3: Measure the Egg
String seed beads onto a piece of thread. You're trying to measure the widest point on the egg, the "waistline" if you will. Keep stringing beads until you have the highest multiple of 5 without the strand being longer the the waistline. This is the length of the longest row. Make sure to write down and/or remember this number.
Step 4: Round Peyote
To go around an egg you need to manipulate this stitch to work in the round. To make a piece of flat round peyote you start with 6 beads and work out in steady increases. This is because of the way that circles and hexagons work together - if you think back to learning to use a compass and making a circle, then aligning the compass on the edge that circle and finding out that 6 circles can align on the edge of the first.
Of course, to cover and egg being able to make a flat circle of peyote is useless. However, if you start with a base of 5 beads instead of 6 you can generally cover something spherical. Depending on the size of the sphere versus the size of the beads it may need to be 4 or 6, so keep that in mind if you are having trouble with this method. Tiny spheres would need 4, huge spheres would need 6.
Step 5: Start at the Bottom
Thread the needle with a doubled length of thread, long enough to be useful but short enough to not tangle.
I've written out the method but the photos are probably more useful for understanding the process.
String on 5 beads. Run the needle through those beads again to make them into a circle. Pull it up snug, leave a few inches of tail (enough tail to put a needle on later.)
On this row, add a bead, go through a bead, and repeat until you've added 5 beads. Go through the first bead you added so that the needle is in the right place for the next row.
Add two beads, go through a bead, repeat to the end.
On this row add a bead in the 'normal' place for peyote and add one in the center of each pair on in the last row.
Step 6: Continue the Increases
You'll want to repeat that same three row pattern - one row of ordinary peyote, one row with pairs of beads and one row with an extra bead in the center of each pair - until you reach the waistline measurement you determined earlier. Only increase at 5 points on each row and keep those increases aligned.
At some point this will start to "cup." Let it. If you fight that you will have threads gapping between the beads.
You will eventually run out of thread. Stop when you have a few inches left. Cut the needle from the thread and take this opportunity to weave in your thread tails from starting. Then string a fresh strand of thread on the needle and tie the ends of the first strand onto the ends of this strand with a square knot. After a few beads the knot will settle in and pull through the beads smoothly. In a few rows trim off any bits of thread that are sticking out.
If you're using very tiny beads, large thread or beads with unusually tiny holes this won't work. In that case, just weave in the tails on one strand and weave through some beads before starting on the next.
Step 7: The Middle Section
This part is kind of coasting - you just do row after row of ordinary peyote stitch for a while. The idea is to keep doing this until you've worked to a point where your row is about 10 beads longer than the diameter of the egg. Put the egg inside the beadwork, and as soon as you think it's safe to start decreasing do so. You want to error to the early side because this can help pull up any slack around the egg.
Step 8: Decreasing to the Top
Decreasing is simple - it's just the opposite of increasing. Track up from the bottom where your increases were, and at those five points around the bead just skip adding a bead - go through two of the beads from the previous row. On the next row place one bead in the gap above the pair that are stitched together. Since the top of an egg is much less like a sphere than the bottom you may have to add a few extra rows in between decrease rows. If you don't like how it's working just undo a few rows and try again. The sphere is mathematically easy to manage, the top of an egg less so.
At the top you'll find yourself with 5 beads. Run the thread through those several times, weave the strand in through some of the beadwork and trim your thread neatly.
That's it! Make a bunch and make a little wire nest, or turn them into a necklace.